You can have your credit card contact their service provider and ask. They are unlikely to tell you for privacy reasons, but they may cooperate with an investigation.

That doesn't mean the person provided accurate information to their service provider, though, and if it's something like a Hotmail or Yahoo address, I think you're out of luck.

If they have their own domain, you can do a WHOIS search and get the name of the folks that registered the domain.