I hesitate to suggest shunning her like the other answers suggest. It seems like it might work, but it also seems like it could be traumatic for her.
A little about autism
I think it's worth knowing a little about how she may see the problem. This is based on my own experience with autism, which probably isn't exactly the same as hers but might give some insight.
To expand on a few comments I've read:
Yes, this is childish behavior. No, it isn't unusual. Autism is a development disorder that slows mental and emotional development. While Alice may appear to be an adult on the surface, she's still a child on the inside, and her behavior makes a lot more sense when you understand that her physical and emotional age are different. Alice is currently inflicting the classic "You're a big meanie so I'm never going to talk to you ever again" punishment on you. The only difference is now her memory is long enough that a nap won't fix the problem.
As someone who's autistic myself, I can easily see how I'd behave the same way if I were a little further along the spectrum. I think the autism aspect is important.
I think it's unfair to expect her to "know better" like everyone else.
- It assumes she's aware she's autistic, which isn't necessarily true. I explained my situation to someone recently and he told me his son is very similar and now he wonders if the son is autistic too. His son is 40 years old. Diagnosis happens late all the time -- especially with females since, as @bigbadmouse points out, they mask their autism better. The OP knows Alice is autistic, but didn't explain whether Alice said that herself or if she just put two and two together and figured it out without Alice's involvement. (Update: Comments say she's aware of her autism, so this won't apply to her, but may to others.)
- It assumes she accepts she's autistic. I spent years in denial myself.
- It assumes she accepts it's a problem. When I hit acceptance, I just shrugged it off and ignored it because I thought it was no big deal.
- It assumes she knows there's help. When I was diagnosed, everyone was so busy fighting that they never bothered doing anything about it. I had no idea help even existed.
- It assumes she accepted that help. There's a good chance I would have rejected any attempts at helping me because I had a hard time trusting anyone and thought I could do it myself. I still do to an extent.
- It assumes the help she got was actually effective. Often, it isn't. I spent two years in therapy making no improvements at all before giving up in frustration.
There are a lot of shaky assumptions behind that assertion, and some potential landmines for taking it.
Being autistic, one of my biggest problems is loneliness. I attend school for several hours a day and church activities twice a week. I even talk to people while I'm there. I still don't have anyone I consider a friend. 
What concerns me is that she may be in the same situation: Even though she goes to social events and talks to your friends, she may still feel left out frequently. It wouldn't be too surprising if your social circle (and maybe some of her catering coworkers) were all the friends she has.
What would you do if your entire social circle suddenly ganged up and started ignoring you? What if you had no idea how you got that social circle in the first place? How would you cope if you couldn't replace them? How would you react "knowing" all your friends hate you now and it's all one person's fault?
If I were in her shoes explaining why I was upset, I might tell you something like this:
I spent a lot of my early life lonely and left out. One day I finally overcame all the anxiety and difficulty and got a job. To my delight, some of my coworkers liked me and made friends. They even invited me to social gatherings! I was in heaven.
Everyone is especially nice to the host. I wanted them to be nice to me too so I could feel liked and important. I watched the hosts carefully for several parties and, when I was feeling especially brave, I announced I was going to host one too. I was very careful to do everything exactly like the other hosts and everything was great! I was so happy!
The first party went so well I hosted another one. I carefully did the exact same thing again to make sure it all went well, and it did. I was doing so well I did it three more times. It was amazing being the center of attention for a change instead of getting ignored all the time.
Everything was going well, so I did it again. I'd been catering events recently and everyone really liked them, so I got the same caterer for my party because they make people like the party. But this time Tinkeringbell said she couldn't come. That wasn't supposed to happen. I double and triple checked everything and made absolutely sure it was all exactly the same as last time (except the improvements), but it didn't work. I was so discouraged I couldn't do anything right and the whole party was a complete failure. Everything went wrong. Nothing was predictable any more.
I didn't get my party and it completely ruined my routine. I was confused, stressed, depressed, and frustrated. I missed my social time, which was really important to me. I slept so much the next week to recover that I missed saying hi to the cute boy at work for two whole days in a row and now I'm scared to talk to him again because he probably forgot who I am by now. Tinkeringbell is the worst person ever.
You'd be hard pressed getting that out of me though. So here are some things to keep in mind:
- Implicit is bad; explicit is good. If you ever write a book, having a character say "I'm angry!" is terrible writing. Unless they're talking to an autistic person. It's a lot easier to process your emotional state if it's made explicit instead of assuming I can pick it out implicitly. I'm bad at subtlety. Especially nonverbal communication.
- Rules are good. I want to know the rules to follow to succeed. I don't care if you think they're insanely complex and situation specific. I want detailed instructions. It's either that or hours of grueling practice with people I'm scared of in situations I'm uncomfortable with. The good news is I can juggle a lot more detail than most (I'm not sure if that's typical of autism) (update: apparently it's very common).
- Communicating is hard. While you're enjoying a relaxing conversation about the latest news, I'm frantically scrambling to keep up with the flood of information you're dumping on me. It would make a huge difference in my life if people actively encouraged me to take it substantially slower than the norm and made sure I knew they liked it when I did.
- Predictability is king. Did you pick that out in the quote above? Many autistic people follow repetitive routines to the point of obsession. Personally, I have to plan small breaks to my daily routine a minimum of several hours in advance and remind myself multiple times or else I'll just go ahead and do what I always do. I might just do what I always do anyway. It's safer that way. Even when I take a risk and do something a little different, sometimes it's so traumatic I never do it again.
Why direct confrontation is bad
As you pointed out, your system for handling conflict is to confront someone and directly ask what's wrong. When people do that to me, it creates an instant conflict. I see other people handle that by responding immediately, and I want to be like other people, so I want to respond immediately too. But my mind is still several seconds to several minutes behind on processing the flood of nonverbal information I just got hit with. My reaction is usually to panic, spit out whatever poorly thought out response pops into my head first, and hope you'll go talk to someone else so I can recover. 
Minutes to hours later, I'll finally develop a well thought out response to your request.
Real-time conversations are hard.
What to do
To be honest, it sounds like you already know why she's upset: You didn't go to her party. Skipping the party may have seemed a perfectly ordinary and reasonable thing to do (everyone does it every now and then, right?), but Alice wasn't prepared to handle it and got hurt as a result. Whether what you did was "justified" from your point of view or not, the appropriate response is still the same: Acknowledge you hurt her, reassure her that you feel bad, try to make it up to her, and take measures to prevent this from happening again in the future.
Since skipping a party hurt her, maybe you can help her feel better by going to one. And being mindful of her situation may give you more success. A less stressful approach would be to give her a letter instead. There's no overwhelming nonverbal information to deal with, and nobody expects letters to be responded to immediately, so she can take her time thinking it through. Something like the following would be good:
I'm sorry I hurt you. I feel very bad and want to make it up to you. If you tell me at least two weeks before your next party so I can save enough money and clear my schedule for you, I promise I'll come this time. --Tinkeringbell
Short, explicit, and to the point. It gives simple, predictable rules too.
Take a little time to make it look nice. Write it on good stationary in a pretty envelope. I'd probably go with plain red or blue, with a flap that stays closed but isn't sealed. Then approach her at the next gathering where she's present (preferably near the end after she's had time to get her socializing in), give her the letter, ask her to read it when she isn't busy, and leave her be.
Give her plenty of time to absorb and think about it and she'll be a lot more likely to give you a well thought out response. Don't pressure her for a response though -- let her give it to you on her own time.
A backup plan
If the letter doesn't work out (and maybe even if it does), you might consider employing your friends' aid in coaxing her away from your social circle. She likes parties for the attention and appreciation, so if you can give her a better alternative, she may pursue that instead (or at least split her efforts and attend fewer parties).
When I was young, I volunteered in my school's library. Every other day, at the exact same time, I'd go in and line up all the books with the edge of the shelf, then put all the books on the sorting cart away. There were no scary surprises, the rules were well understood, and following them reliably got me appreciation from the librarian. Working in the library is still one of my best memories of school.
Some places (like libraries) rely heavily on volunteers to help with their routine maintenance. Many of them "pay" their volunteers with heaps of praise and appreciation. If you or your friends can (and are willing to expend the effort to) get her involved with one (or better, several) of them, she'll likely feel at home in an environment with easy rules that reliably get her the attention she wants. It'll probably appeal to her more than the parties and distract her from stalking you all the time. Inviting her to join you on visits to them a few times and then assuring her she can go on her own too would likely work well.
Having multiple active social circles would also be a big improvement to her emotional health. It'll be easier to keep going and seek out replacements for unhealthy circles if she can turn to several others for support in the meantime.
Next time (if she gives you another chance), you can try to be sensitive to the fact that there's a reason for her parties: they're a bid for attention; an attempt to fit in with and be liked by people she desperately needs but who frequently hurt and confuse her. That doesn't mean you always have to go, but at least reassure her that you're happy she invited you and appreciate her caring for you. Make sure she gets some success, even if it doesn't go exactly according to plan.
Also, it may be worth introducing her to some other easy ways to get attention that are less likely to be rejected. Bringing baked goods to social gatherings, for example. The investment needed to take a cookie and say thanks is low, so she'll be likely to succeed often. Teaching her a variety of strategies and encouraging her to pick a different one each time will help keep her from getting too repetitive.
@JarkoDubbeldam really described the danger of shunning nicely in one of the comments:
From my own experience, negative experiences stick around for way longer than positives and can easily overwhelm them. I personally find it really hard to forgive. The negative experience would create a reaction, and in future references to that negative experience (in this example you might be that reference) it is very easy to default to that same reaction, to that same feeling. Even if at some point the default is not at all appropriate anymore. The default "rule" is just so baked in that it becomes hard to step away from it. Even to admit that it might not be appropriate.
Do note that I am not condoning the behaviour, but rather give a point of view as to why someone might act that way.
-- Jarko Dubbeldam
This is a combination of the Recency Effect and a Self-Fulfilling Prophecy. People's most recent experience with something has a disproportionate weight in their overall impression, so when the most recent experience is bad, it taints the whole impression. Then they expect the next encounter to be bad, causing them to be paranoid about every little problem, resulting in it actually becoming bad simply because they expected it to. Bad last impressions kill a lot of relationships.
Being often confused and overwhelmed by the world have given me a lot of insecurity, which makes the risk of a bad last impression a common experience for me. When the rules as I perceive them suddenly change out from under me (a regular occurrence), the negative experience has a strong tendency to make me just avoid the situation rather than trying again. It's better to stick to what's predictable and safe.
I think that ideally, she should be alerted to the change as early as possible and given the new "rule" so she can adjust to it. Easing into the adjustment doesn't always work, but it helps. For the first time,
Alice, I get upset when you interrupt my conversation with Tinkeringbell. I've been patient a long time because I like you and I thought you'd forgive her and be friends again, but you've been like this a long time and I'm starting to get frustrated. If you don't want to share a conversation, please wait until I'm done and I'll be happy to talk to you next.
Use that once per friend, then skip it afterward.
Note the explanation for why Alice was allowed to break the rule in the past (the speaker likes her and hoped she'd forgive) and why that's changing now (she hasn't forgiven for a long time and the speaker is frustrated). Framing it as an extension of the rules she already knows instead of revoking them shows her they haven't changed, and will maybe give her some hope they can still be understood.
Making it clear the speaker still likes her and wants to talk to her too is also important. It helps her know she hasn't completely ruined things yet and there's still hope for recovery.
The next time,
Alice, if you don't want to share the conversation, please wait until I'm done and I'll be happy to talk to you next.
Alice, you're starting to frustrate me. If you keep interrupting, I'm going to stop paying attention to you until I'm done. I'll give you a turn after I'm done talking to Tinkeringbell.
Then ignore any more attempts.
She might guess that this applies to every friend after hearing it from the first one, but I'd give her the benefit of a doubt and give her the first warning from each friend.
The risk here is that she'll be discouraged by the rejection, leave the party and avoid the rejecting friend (or the whole group) after this. It will likely help her a lot if someone else she's still friends with intercepts her quickly and invites her to talk with them instead, since it would reassure her someone still likes her. It would also help a lot if the rejecting friend specifically seeks Alice out after talking to you and asks her what she wanted to say, so she knows the rejection was temporary and the rejecting friend still wants to have her company.
She might give terse answers due to persisting ill feelings, but just keep probing and showing interest until she opens up or gives a clear rejection.
Positive reinforcement of good behavior would be a good idea too. It's always encouraging to hear you did the right thing, even if it was hard or you didn't know whether it would be good. During the approach, say sorry for making her wait and thank her for being patient when you go find her again.
I wouldn't go to her parents behind her back as some have suggested. You have no idea what her relationship with them is. Personally, I like my family a lot, but I'm extremely insecure about discussing my autism with them. If you brought it up with my parents, I'd be extremely embarrassed and pretty mad.
If you can pull it off, a better idea would be to organize a very small party with you, her, and a few close friends. Make it a "Parent Party". Tell everyone to bring a parent or two (she might not have two). Provide plenty of details about places, times, and what you're planning on doing (something good for socializing around, of course). Make sure everyone seems to be looking forward to the special party and inviting their parents. If she's comfortable mixing her parents and friends, she'll probably imitate everyone else and bring a parent (or two) too.
If you're lucky, her parents will have seen her behavior toward you a few times already, pick it out right away and help you resolve it even before being asked.
@Dzyann pointed out that even once you get back on Alice's good side, this will probably happen again the next time you have to skip one of her parties. That puts you under a lot of pressure and sets her up for failure.
Setting some ground rules would likely help. In the letter above, for example, I asked for two weeks notice to have plenty of time to rearrange schedules.
Since Alice went through a period of time where she couldn't afford to attend parties, she's likely to understand when you explain that now you're going through a period where you don't have a lot of money, and can't afford to attend parties either. It's possible she might decide to solve the problem by offering to pay your part the way you helped pay hers.
Another good approach would be to invite her to help in planning your parties and those of your friends. Chances are she thinks the rest of you are at least somewhat "better" at this than her, so offering to let her watch how you do it may appeal to her. If she doesn't think you're better, she might find it appealing anyway since she seems to enjoy parties. She'll be a lot less likely to completely ignore you if it means losing out on chances for fun activities together.
People also tend to hide their mistakes, so it's easy for autistics to get the impression that everyone else is perfect and never fails at anything. By letting her see you try, fail, express frustration and recover while hosting your parties or doing other activities, she'll get the more realistic view that setbacks are normal. She'll also get to see recovery systems in action and maybe start employing those in her own parties for higher success.
 I consider everyone I know to be acquaintances since we mainly discuss school work and never go out to do anything together. I've never had anyone I consider a friend.
 Autism is one of those things that gets better with time. Eventually the "normal" people fully mature and essentially stop developing mentally and emotionally, giving the autistic ones a chance to catch up. It's easy to tell when a ten-year-old is acting five, but a lot harder to recognize a forty-year-old acting twenty. It might take a while depending on the severity of their autism, but they do improve a lot over their lives. Someone suggested to me that a good rule of thumb is that autistics' emotional age tends to be about 2/3 of their physical age.
 I once told someone I was wearing a black shirt today because I couldn't find anything but hunting shirts, with the implication that I checked every store in the whole city. I did check every store in the city (and 500 pages of Amazon search results), but I found a lot more than hunting shirts. I'm honestly surprised he took that the way he did -- maybe he wasn't paying attention.
 People are also heavily biased toward the first impression (the primacy effect). Overall impressions are generally made up of a first impression, a last impression, and everything else. Since "everything else" is a much larger category, individual experiences in it tend to have a disproportionately low weight.