You can order a Thanksgiving dinner from the resort to be delivered to your room. You should get something in the mail from the resort a few weeks ahead of time, with ordering information. They usually also send a list of activities for Thanksgiving week.
Or, you can order a dinner from one of the grocery stores, or prepare it in your villa.
As far as restaurants, we've never gone out for Thanksgiving dinner -- but I know the major hotels usually offer Thanksgiving dinner buffets; many of the restaurants are open, but I'm not sure how many of them offer traditional turkey dinners, if that's what you're looking for.
DH and DS like the Indigo Run golf course a lot, as well as any of the three at Palmetto Dunes or at Shipyard plantation. They're all fairly close to the Disney resort and should have reduced green fees at this time of the year.
You will LOVE Hilton Head for Thanksgiving!!
