Re: vegetarian at Disney?
Assuming they have made ADRs then they could have notated the fact that they were vegetarians. However, even if they have not notated it, when they go into the sit down restaurants they should immediately notify their server that they are vegetarians and of their dietary restrictions. Disney are extremely good in watching out for cross contamination of their food. Almost every TS restaurant has at least one vegetarian item on the menu and I feel relatively certain that their chefs could fix up something special for you after they come out and talk to you.
No as for CS places, I know there are vegetarian options but I'm not sure if you can guarantee that they have not been cross contaminated.