I can't speak to the non Disney owned restaurants and food allergies, but there was a story on this week's Disney Dish podcast about Takumi-Tei restaurant's so-called plant-based menu items at the Japanese pavilion.
The Disney website menu for this location states: Our plant-based menu items are made without animal meat, dairy, eggs and honey. Someone who is vegan ate there recently and there were multiple instances of non-vegan items (such as egg yolk, honey, real cheese, and even duck) in the plant-based menu. When she asked the chef about it was clear they didn't understand the difference between vegan and vegetarian.
Sorry, I know vegan/vegetarian is not your concern but this story is to point out you may need to be very specific in your questions to the chef before ordering at any Disney restaurant.