Hello Mandy. Yes, I believe many of the restaurants do serve Christmas Day menus. If this is an important factor for you and your family, when you make your Advance Dining Reservations, you can ask for further information on this.
I can speak to the one restaurant we have dined at specifically on Christmas Day, the
Liberty Tree Tavern at the Magic Kingdom. We had a terrific Christmas meal, with all the traditional fixens. If I recall, the food included turkey, ham and more. Another consideration would be some of the fine restaurants that are within the Epcot World Showcase. We have celebrated other holidays within the United Kingdom, including at the
Rose & Crown Dining Room and
Le Cellier Steakhouse. You won't be disappointed with the food at either of these locations.
Hope this helps and enjoy your trip!