Anyone running an online store is occasionally going to get a "customer" demanding a refund when there is in fact no record of a purchase. Following is the routine way of dealing with such:
1. Check not only your store admin orders, but also your merchant services accounts for any record of the transaction.
2. If you don't find it, let them know you haven't been able to find any record of the transaction so you'll need to verify it before you can provide a refund. Ask them to send you evidence / specifics on the transaction so you can verify.
3. Then double check to see if you can verify if the money was deposited to your account.
Usually these kinds of things are an effort to rip you off. Whatever is going on, following these steps should handle the situation.
If there was an actual transaction, by now you would have been able to verify it.
Maybe (we've had this happen with clients) they were confused and it was a different store they had made the purchase from. If they haven't realized it by now, you'll have the evidence to point out to them ("the charge was from XYZ Company not us, ABC Company").
If it is a rip-off, they usually won't respond at all, since if they send you a phony bank record or some such, they would definitely be committing attempted fraud at that point.
They might just get mad and repeat their demand, in which case you just patiently repeat "company policy prevents issuing a refund without verification of the original charge" until they either provide the evidence or go away.
It is important in cases like this to be patient and non-confrontational. That defuses the (rare) case where you just have a crazy person on the other end. You don't want them to go away mad, you just want them to go away.