We only do counter service and my daughter (allergic to milk, egg, peanut and sesame)!eats better and safer there than anywhere except home.
We eat a Cocina Cucamonga (sp?) a lot. They are peanut-free. We also have eaten at Tomorrowland Terrace, French Quarter, and Smokejumper's Grill without problems. Some of the places now have allergen menus, but since we are dealing with multiple allergies, we always have to talk to the chef. For treats, popcorn and sorbet (with an allergen note and special precautions taken - I would think that other scoop shops may be able to do the same. We just haven't had the option because of her dairy allergy) from Ghiradeli. Dole Whip should be fine, too (but my daughter is young, and pineapple gives her diaper rash). We also sometimes get a vanilla cream frappacino from Starbucks (hers made with coconut milk) as a milkshake treat when the other girls have ice cream.
I'm sure you have more options than this, but this is what we have found are safe for all her allergens.