Kayle:
So glad you enjoyed your trip! My husband and I always tend to plan our next trip to Disney on the road home! So many new things to discover and places we want to spend more time at!
The two restaurants that come to mind when you talk about food and games are
Toy Story Pizza Planet at Disney Studios and the
ESPN Club on the Boardwalk. (I don't recall any place at the Magic Kingdom where you can play games and eat, unless you grab a chicken leg in Adventureland and walk over to the shooting gallery!)
Toy Story Pizza Planet is probably closest to what you have in mind: They offer personal pan pizzas (which are just o.k. in my book) and an arcade full of games. Just like Chuck E Cheese, it can get LOUD, so find yourself a table tucked away upstairs!
The ESPN Club is "grown-up" alternative similar to a restaurant in my neck of the woods called Buffalo Wild Wings, but on steroids (my 9- and 11-year-old sons love it.) If you've got older children, you may want to consider it as you have arcade games you can play at your table as well as those in an adjoining room! It's a favorite of Disney Dad Doug. Feel free to shoot him a question about it through his profile page!
Hoping July gets here real soon for y'all!
Whitney