This is what it says on the cake thread on the top of this page, but maybe ask in there.
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What if someone in my party has dietary restrictions?
In general, Disney will do its best to accommodate dietary restrictions like allergies, but it’s best to
call to find out your options. Usually they will refer you to vegan bakery
Erin McKenna’s Bakery (née Babycakes NYC) at Downtown Disney, which delivers two-layer 6-inch cakes (serves 8–10) to restaurants on Disney property for $52. The price of the cake is $59 if you pick it up at Erin McKenna's.
Call 407-938-9044 to order (Monday – Friday, 9am–5pm). Erin McKenna’s offers gluten-free, wheat-free, soy-free, casein-free, egg-free, vegan and kosher baked goods. Certain other Erin McKenna’s Bakery items—like mini cupcakes, cookies and doughnuts—can be found in the gift shops at the Beach Club and Kidani Village; at Jambo House’s quick-service spot, the Mara; in the food courts at Pop Century and Art of Animation; and at Animal Kingdom’s
Gardens Kiosk, a food-allergy info booth near Flame Tree Barbecue and It’s Tough to Be a Bug."