All the restaurants and counter service places have kids menus. IF you are at a table service, the servers will work with you to find something your child can eat. Also remember that if there is a side dish anywhere on the menu (like fries, rice, baked potato) listed with a meal, you can ask just for the side.
At counter service, if the kids menus don't appeal to you and there is something like a basket with fries in it, you can always ask for just the fries.
Don't think you have to get the entire meal as it's advertised. My husband will often substitute fries for rice even if the menu description lists his entree as coming with fries. Or the other way around.
If nothing more, there's McDonald's (not called a McDonald's but they are there) in each of the four parks for chicken nuggets and fries (no burgers).
There's really no reason for anyone to go hungry at Disney World.