There's no way to try to talk to her without risking annoying her or being perceived as rude.
However, life is full of risks and inaction will probably just lead to her playing on your mind for weeks on end before something happens and your chance is gone for good. So instead of 'no', I'm going to say don't do nothing.
There's a very simple thing you can do to discover if she is interested in striking up some kind of rapport, but be prepared to accept that she might not want to.
- Make sure you don't have a drink.
- Wait until she's finished her coffee
- Go over to her and wave (or do some other gesture) to get her attention
- Tell her that you were about to get a coffee and ask her if she'd like you to get her one since you're getting up
If she says no, just go get a coffee for yourself, then forget about her and move on.
If she says yes, buy both your cofee and her coffee, bring her coffee to her and then go back to your seat and get back to what you're doing.
If she offers to pay or wonders why you did it then say you 'just felt like being neighbourly' or 'just felt felt like doing a good deed'.
If she's interested in having a conversation or becoming friends she'll probably talk to you or return the favour in the following days and you can take it from there.
If she doesn't attempt to make conversation or return the favour in the next few days, just forget about her and move on.
(Possibly even move away from her if it helps.)
It might seem hard to let go, but remember that you don't actually know her, until you've spoken to her and got to know her anything you think you know about her is just a story in your head.
(Also there's not exactly a shortage of humans, if she's not interested you'll find another one.)
In summary, the worst case scenario is that she's not interested, but it's better to be reasonably sure than to assume and never really be sure or to sit there wondering until it's too late.
Even if she does want to become acquainted, please remember that's still a long way from being ready for a relationship. It might be the case that she'll only ever want to be friends or that she's happy to exchange coffee purchases now and again but doesn't want to become more familiar than that, in which case you either have to accept that or move on. I honestly hope it goes amicably, but you should always be prepared for the worst case scenario, which in this case means letting go.
Lastly, I acknowledge that there are some flaws in this.
For example, she might refuse a drink because she might think you're going to spike it with drugs or something. (Though if that's the case then you'd probably have had a tough time getting to know her though other means anyway.)
Trying to make contact with strangers is almost always going to have its difficulties. Generally easier to meet people through things like clubs, where you're expected to talk to each other and are likely to have a common interest.