ok, here goes - for more detail, please go to the 'restaurant review' sticky and read my posts!
counter service
tomato soup many places
mushroom croque at Victoria's
cheese baguette at the train station
ditto at New York sandwich place but it was AWFUL
table service
buffets at SQL and NPB were not veggie friendly and had a sort of 'tough luck' attitude - don't recommend them
Walt's had a great veggie appy, entree not appealling but they would serve double portion of appy if requested
Auberge de Cendrillion was great; lots of choice on the appy buffet
Silver Spur felt that anchovies are just tiny fish so they don't count; they did have an excellent tomato stack appy and a very unusual veggie cottage pie entree
Blue Lagoon - I had a lacklustre cold bulger appy and no entree but made up for it with chocolate lava cake!
King Ludwigs - lots of choice, excellent food, one of the best choices!
HNY had a few choices too but I didn't make it over there. Generally, a lot easier to find veggie food, in particular at counter service, than in the US parks. Check out www.dlpfoodguide.com for sample menus.
Also note that in France veggie is used very loosely, as in my anchovy example. Make sure that you are not getting fish, or chicken broth, etc in your 'vegetarian' entree.
If you look at www.dlpfoodguide.com that will give you an idea of what each eating place offers. My sister is a vegetarian and found the choice limited in places.