Very a very careful and non-invasive way, she probably gets a dozen requests a day. Make some small talk every time you go to the store, If she's not flirting with you or doesn't remember you I'd forget it. If the flirting is going well I'd ask for her email address since as I said she surely gets several guys a day asking for her number.
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Approach the cashier girl when she is not busy, make friendly conversation, and then ask her if she would like to go out for coffee or a meal sometime. Be polite, respectful, and mindful of her response.