Hi, Jenelle! While there are not locations that are specifically vegan or gluten free, you will find options that fit both special dietary needs throughout the Magic Kingdom.
Counter service vegan offerings include French onion soup without the cheese at Be Our Guest, salads at various locations, and garden burgers at several counter service stops. You'll also find fresh fruit, vegetable burrito, and even Tofutti "ice cream" dessert.
If you want a table service dining experience, Crystal Palace has some wonderful salads and at least one vegetarian option on the buffet. You can also talk to a member of the culinary team to see if there are other options they can adapt for you. Liberty Tree Tavern lunch has a quinoa burger, and the Plaza Restaurant has a veggie sandwich. You can also find vegan options at Tony's Town Square, which features Italian cuisine.
Gluten free options are pretty widely available at all of the theme parks. There are breads, pastries, and pastas available that are gluten free, as well as some sweets and snacks. Walt Disney World is awesome when it comes to
Special Dietary Requests. Gluten free is one of those specific dietary needs that is pretty well recognized and supported at Walt Disney World.
I hope you have a great time eating around the Magic Kingdom!