Assuming that the menus on AllEars are correct here are some of the options available for childrens appetizers besides raw veggies:
Hollywood Studios
50s Prime Time: Salad or Chicken Noodle Soup
Brown Derby:Chicken Noodle Soup or Chopped Salad
Mama Melrose:Garden Salad or Vegetable Minestrone Soup
Sci Fi:Tomato Soup or Garden
Epcot
Coral Reef:Soup of the Day
Chefs des France:Cheese Puff Pastry or Mixed Green Salad
Le Cellier:Canadian Cheddar Cheese Soup
Nine Dragons

ot Stickers or Vegetable Spring Rolls
Rose and Crown

otato Leek Soup
Tutto Italia:Fruit Cocktail, Strawberry Salad, Mozzarella Sticks (veggies sticks not a specified option here)
Magic Kingdom
Liberty Tree Tavern: Garden Salad or Chicken Noodle Soup
Plaza:Tomato Soup or Mixed Salad
Tonys Town Square: Garden Salad
Resorts
Sanaa:Mixed fruit, Tomato soup with cheese toast
Kouzzina:Chicken Noodle Soup or Garden Salad
Shutters:Garden Salad or Chicken Noodle Soup
Olivias:Garden Salad or Chicken Noodle Soup
Kona Cafe:Garden Salad or Chicken Noodle Soup
Boatwrights:Garden Salad or Chicken Noodle Soup
Turf Club:Garden Salad or Chicken Noodle Soup
Whispering Canyon Café:Fruit Cup or Garden Salad
Captains Grill: Garden Salad or Chicken Noodle Soup
DTD
Captain Jacks:Mixed Green Salad or Soup of the Day
In the past, we've gotten the puff pasty at Chefs des France and it's quite large. Both mom and dad were able to have a taste. It was delicious.
Also, we've shared the lobster bisque from Coral Reef with one of the kiddies and the cheddar cheese soup from Le Cellier.
Just a few suggestions!!