At the Animal Kingdom, there aren't very many table-service options, but the Rainforest Cafe should be entertaining for all of the children with all the animals, rainforest theme, etc. We enjoy the Flame Tree BBQ, which is a counter-service restaurant at AK. All seating is outside, but there are lots of quiet nooks and crannies so babies can sleep in the shade. Follow the path all the way down to the water and you can enjoy lunch and a fabulous view of Expedition Everest, and sometimes a boat goes by with Disney characters, which is very exciting!
As far as other restaurants at resorts, many have buffets so you can eat quickly. The Whispering Canyon Cafe at the Wilderness Lodge is one of our favorite places to take our toddler. You do order off the menu, but there is a family-style "skillet" option with shrimp, ribs, chicken, sausage, baked beans, corn on the cob, and mashed potatoes with cornbread. Throughout the meal, the waitstaff engages the kids in games. Very loud but always fun!
Boma at Animal Kingdom Lodge has an amazing buffet with many choices, and we enjoy eating there with our toddler as well. There are several character buffets at the resorts that you can also look into based on your 6 year old's interests.
Have fun!