Hi, Jennifer! I have always found Walt Disney World to be one of the most allergy friendly places. We've visited with people with various food allergies or dietary requirements, and I've never had anyone give us a sideways glance at Walt Disney World. They do their very best to accommodate special dietary requests.
You will find either Rice Dream or Tofutti ice cream in each of the theme parks. The Plaza Inn Ice Cream at the Magic Kingdom and Sunshine Seasons food court at Epcot both have dairy free options. At Disney's Hollywood Studios, Hollywood Scoops has Tofutti Frozen Dessert, and Disney's Animal Kingdom offers both the Rice Dream and Tofutti.
OMG Manufacturers and Enjoy Life products are both available throughout the theme parks and resorts. You'll find cookies, brownies, and cakes, and you can usually find some type of candy that will work for someone with these allergies.
For more help with special dietary concerns, check out
Disney World Special Dietary Request. This makes planning much easier when visiting with someone with allergies.
Have a wonderful vacation.