Well I don't know what your children's tastes are but I will guesstimate with what I am familiar with when attempting to feed my 2.5 year old nephew which can be difficult because he's a picky eater and besides breakfast he doesn't eat much so he tends to share with my sister. These counter service restaurants are the places we enjoy so I'll list what we've ordered for him.
I will omit Chicken Nuggets, Hotdogs, Mac & Cheese and Pizza which can be found just about anywhere (if it were up to him he would eat these items all the time if it weren't for us being adamant about him trying new things)
MK - Columbia Harbour House (Kids Chicken & Fish Basket), Cosmic Ray's (1/2 Rotisserie Chicken meal shared w/ his mom)
Sleepy Hollow (Waffle), Tortuga Tavern (Barbecue Chicken Sandwich shared w/his mom)
Epcot - FW: Sunshine Season's (Breakfast: French Toast, Lunch: Pizza Panini or Chicken Drumstick)
WS: Boulangerie Patisserie les halles (Ham & Cheese Croissant, Quiche Lorraine, Yogurt Parfait w/Berries), Katsura Grill (Teriyaki Chicken), Liberty Inn (Grilled Chicken Plate), Sommerfest (Apple Studel, Nudel Gratin), Yorkshire County Fish Shop (Fish & Chips)
DHS - Backlot Express (Dark Side Chicken and Waffles), Fairfax Fare (sheared 1/2 Chicken w/his mom, Banana Parfait)