Possibly because you can go to the store and they should have it in their records. They might tell you that they can't show you but if you know the credit card number, or you can recite your name and zip code, bla bla bla to prove it's you, (or, if you want, get the police involved) and try if that can track them down. BUt you should just cancel your credit card in the meantime, because it's doubtful you can track THEM down when it's YOUR card.
Depending on the state you live in you might get a break! If you report it right away some states hold you liable for the first $50.00. The longer you wait the more liability you suffer! Yes the cops can track the sales and with the video in the store can make a good case!
If they use the credit card in stores and they take the time to view surveillance cameras then yes they could find the person, if they use the card online then good luck. Been there and nobody wanted to bother with it.