You will have no problem ordering from the kids' menu at any counter service restaurant and you will be able to order from the children's menu at virtually every table service restaurant where you order from a menu. Once in a while someone will post that a server or restaurant adhered to the ages 3 - 9 at a table service venue, but not often.
You will charged the adult price at buffets (e.g. Tusker House, Crystal Palace) and all you care to eat family style restaurants (e.g. Garden Grill, 'Ohana), fixed price meals (e.g. CRT, Akershus), dinner shows (Hoop De Doo, Spirit of Aloha Luau, Mickey's Backyard Barbecue), dining packages (e.g. ROL, Fantasmic), special events (e.g. fireworks dessert parties, Dine with an Imagineer) and the family platters at Whispering Canyon Cafe and Liberty Tree Tavern lunch.