I don't know how big your party is, but I just checked thanksgiving day for 4 people, and as far as Magic Kingdom, or not too far from there, I came up with Tony's Town Square (Italian) at 11:45 a.m. in MK; Plaza at 1:05 in MK (mostly sandwiches if I remember right)
For nearby places there was also one available at 11:15 a.m. at the Trial's End at Fort Wilderness. It is a buffet and it will have thanksgiving type food that day. Personally, if it were me, I would jump on that one. You can take a boat there from MK. The only problem is that if MK closes to capacity that day, you may not get back into MK after eating.
Since you said "afternoon", I also ran a seach on anything available from 12-4 anywhere at WDW. There are only 21 TS places anywhere on property with availability in that time frame. None in MK, or even close. Mostly they are DTD or Epcot WS places. There are a couple at the resorts. Most of them are places I don't think you will get the traditional American Thanksgiving. Kouzzina over at the Boardwalk had a 2:15 p.m. I've heard that it specially serves traditional thanksgiving food that day, but I don't know that for a fact.