The local delicacy is conch (pronounced conk). It's basically a big shellfish which is a bit like squid, but you can get it cooked in a variety of ways. We had conch fritters which were excellent.
Locals dry it and cure it so they can eat it throughout the year rather than just when it's in season, however on Provo there is a conch farm that you can visit so there is always a plentiful supply.