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Three of the 5 charters I use most commonly in Boynton Beach, West Palm, and Jupiter are open on Thanksgiving. The other two are taking the holiday off. Many of the operations have an online calendar that makes it very easy to check for availability of charters.
Aqualife and South FL Diving HQ in Pompano, Sea Experience in Ft. Lauderdale, Horizon and Rainbow Reef in Key Largo are all running charters on Thanksgiving day. I don't dive in Key West so I didn't check there but they are most likely running as well.
All this info is available online. Just about every charter has an up-to-date website to see when they are running and what their availability is. Most only take Christmas Day off and I've even seen some run charters on Christmas. Money talks.