The rate is set by the card issuer and agreed to by the borrower. The other type of card is a variable rate credit card. The rate charged for outstanding balances will be tied to the prime rate. Vince should see any rate drops almost immediately. He can check his original credit card agreement, his statement or call the card issuer.
Most cards are variable, but the only way to know for sure is to check.
Vince should transfer any open balances to variable rate cards. He'll also want to watch his statement to make sure that the rate has been lowered. That should happen automatically. But, if he doesn't see a lower rate in the next billing cycle or two, it wouldn't be a bad idea to call the card issuer to find out why the rate hasn't changed. No matter what's happening with the rates, Vince can always try to get a lower rate by phoning the card issuer.