First, a little frame shifting
You want to court her. Assenting to courtship is an act of whimsy. What are the prerequisites of whimsy?
I must disclose agenda: I want a world where women are inherently safe and empowered. Hardly altruism; that works in mens' favor too. Point is, right now they don't, and men "doing our usual" isn't going to move the needle.
Back to the prerequisites of whimsy. As it happens, safe is the first one. Can't be whimsical if you're worried about life or food or shelter or someone who seems a threat. You can guess empowered is too, but there's a third: ability. If she can't fit dating into her schedule or budget, if family or social or work issues prevent her (can she date customers?) then nope.
Special issue: when one is working, one has to do ones job or get fired. Worse the job requirements often entail being nice to customers, and definitely engaging customers and remaining at ones post. That fact can confuse the romantic, so special handling is required.
This is also a longshot. Probably about a 95% chance she either is already partnered or not looking, for some good reasons you can't override. On top of all the other overlapping chances of something else being in the way.
It's also good to sanity-check your own motivations. Make sure you aren't unconsciously seeking self-validation of some kind. If "no" seems like it would damage, hurt or invalidate you, get to healthcare and work that out. That is a requirement of the principle of safe -- which also applies to you. And there's the factor that when men are hurt, they sometimes hurt back. Her having to worry about that violates the principle of empowered.
Since it's about whimsy, the goal here isn't "yes" - the goal here is "why not?"
So we can cross off several wrong approaches
Giving away ones phone number is a safety risk for the girl with indeterminate reward - silly to ask. A phone number may be voluntarily given later, but should be asked for by saying "hey, I'd like to stay in touch with you" and let her choose out of the variety of contact methods that exist today.
Asking while she is serving you as an employee violates empowered because she is effectively cornered: she can't step away, she has to serve you. There's no freedom of motion for her.
Too direct an approach also creates a Huge Awkwardness when you visit the shop in the future, as now she must explain her rejection. Awkwardness is the enemy of whimsy. That may not have occurred to you, but like I say, "men doing our usual" doesn't create any safety or empowerment. Any approach must provide an easy exit for her so there'll be no uncomfortable conversations next time.
Again, not altruism: it's in our interest; the path to "why not?" isn't found by making her uncomfortable.
The open door
Since your hunch is that she does like you, all you really need to do is open a door. This doesn't require breaking any of these rules, but it requires finesse. Consider a series of improving options.
Hi. My band's playing at the Speakeasy this Thursday, and I wanted to let you know about it. (gives performance flyer)
By the way, having this sort of opening is a big reason boys start rock bands. Anyway, it doesn't have to be a rock band, it could be a sporting match, play, whatever - just as long as it's a small enough venue (Taylor Swift concert is much too large) and the entry cost isn't too high. It has to be something any random girl would find enjoyable all by itself even if you weren't involved.
But we're still not clear of the problem of putting her on the spot. She may feel like she has to accept or decline right there. She will expect you, on next visit, to quiz her about "did you come?" or "Did you enjoy it" or even worse, "Where were you?"
I mean, aside from the 95% factor, she may have a prior engagement that night, so her failure to appear is not even a rejection, and she doesn't want to worry about you feeling like it was.
Oh sorry, I'm a little distracted, I'm trying to get in the zone. My band is playing at the Speakeasy Thursday night and it'll be our first performance all year.
Much better, now we've avoided putting her on the spot, we've just slipped her a little bit of intelligence, with no request whatsoever for her to act. She might followup with other questions like "what time?" and in that case, don't answer in a way that creates the expectation. She will appreciate you phrasing it in a way that doesn't put her on the spot.
The flaw in this approach is there isn't really an explainable reason for telling her that. I don't think it'll make her uncomfortable, but there's a finer way still.
Who do I talk to about putting something on the store's bulletin board? My band is playing at the Speakeasy Thursday night, we're so excited."
Fair chance you'll get a "Oh, I'll put it up for you" or "We don't have one, but I'll put it in the staff break room". Here, you're just leaving a breadcrumb trail. If she's interested, she'll follow it unless unable. And it's low-impact enough you can do something like it several times without any appearance of being creepy.