In my experience (I'm lactose-intolerant and DS has a milk protein allergy), most table service restaurants have either sorbet or dairy-free ice cream (usually tofutti or rice dream), even if it's not listed on the menu. For the ones that don't, they will have some form of dairy-free dessert option (sometimes labelled on the menu as plant-based). You can also find dairy-free ice cream, sorbet, and other desserts at several other places throughout the parks:
In MK- Plaza ice cream parlor has dairy-free icecream and sorbet. Aloha Isle, Storybook Treats, and Sunshine Tree Terrace all have Dole Whip and Dole Whip floats (no dairy in Dole Whip).
In Epcot- Kabuki Cafe has Kakigori (shaved ice). L'Artisan Des Glaces has sorbet.
In HS- Hollywood Scoops has dairy-free icecream. The blue and green milks in the Star Wars area are dairy-free. Tune-In Lounge has non-dairy ice cream and milkshakes. Harambe Market has shaved ice. Tamu Tamu has Dole Whip.
Mickey pretzels (without the cheese cup), popcorn, strawberry fruit bars, frozen lemonades and slushies can all be found throughout the parks as well and are all dairy-free.
