I would think that they would already have to, don't regular businesses have to keep track of that sort of thing when they buy from their supplier?
They do keep the sellers information and it is actually kept in a digital database required by the state. That being said it is only a name and an address, and like 99% of criminals the address on their drivers license isn't where they are living. How do you fix the problem? I don't have a good solution for you. Maybe a 5 day hold before cash is given, or something a long those lines. "Usually" that is the time frame an item is pawned before it is located. Most victims don't realize they are victims for a few days, until they go to grab that item.... and hey its missing.
How do you solve this problem for good? Get rid of pawn shops, is there a legitimate use for them? sure? is probably half of there business stolen crap? dunno but its probably pretty close.
If you got rid of pawn shops property crime would drop big time. What drives property crime? Drug use. Could there be consequences? Possibly, it could mean more robberies, purse snatches.... etc.
I don't have the ultimate solution, I do know one thing criminals piss me off, that's why I do what I do