We have a very extensive list of allergies with our 3 boys and it's not uncommon that chefs have to go off menu to prepare something safe for them. We have been able to get meals at pretty much all Table Service restaurants but struggle with quick service. For just dairy/soy though you should be able to find more QS options as well.
The ones that have really stood out for us in the past as going above and beyond are Crystal Palace and Biergarten.
But we've had good experiences at Raglan Road, Be Our Guest, Cape May Cafe breakfast, Hollywood Brown Derby, Sci Fi Diner, Cinderella's Royal Table, Rose & Crown, Captain's Grille, The Wave, Le Cellier, 1900 Park Fair, Chef Mickeys, Hollywood and Vine, Rainforest Cafe, Skipper Canteen, Liberty Tree Tavern, Tusker House.....and I might be missing one or two.
We had "iffy" experiences with T Rex, Ohana, and 50s Prime Time. Hit or miss with the chefs and/or having ingredients that were not already prepped with allergens. But again that's with a long list of allergens. Dairy/Soy should be relatively easy to work with.