Hi Jennifer,
It sounds as though a lunch or dinner buffet would suit your family best and there are some great ones in Walt Disney World. One of my families favourites is the Crystal Palace in the Magic Kingdom, the variety of food is huge including shrimp, rotisserie chicken, prime rib, pastas, salads, kids foods and delicious desserts. Another reason my children like it here is because Winnie the Pooh, Eeyore, Tigger and Piglet are there.
Another good choice is dinner at 1900 Park Fare at the Grand Floridian. This is also a character buffet with Cinderella, Prince Charming, Lady Tremaine and the Step-Sisters. The food is quite similar to the Crystal palace but seemed to me to be a little bit nicer!
What I suggest you do is have a look at the dining pages of the website
www.allears.net here you'll be able to view all the menus and see if any suit your families needs.
Hope that helps and you have a great trip,
Jo