I had a really good veggie sandwich at Roaring Forks CS at the WL.
I agree with much of the above, but I'll list my favs (and not so favs).
Epcot
Favs
Sunshine Seasons CS has a lot of veggie options.
Tangerine Cafe CS in Morrocco has a great veggie plate (hummus - yum).
Le Cellier TS has only one veg dish at a time. I've eaten there twice, and both times, the chef is willing to modify to your tastes and both dishes were FABULOUS.
Not Favs
Skip Garden Grill. In fact, I'd skip any family style place. I just didn't get the impression that they would work with me and I didn't get any choices or opportunity to have a choice.
MK
Fav
Columbia Harbor House CS Lighthouse Sandwich - yum!
Crystal Palace TS - dinner buffet had many veg choices.
Other
Jiko - also only had one veg dish, and I wasn't a fan (probably more of a taste thing and the veg option has since changed). However, there were two veg appetizers that we tried and those were delicious!
Check the menus
http://allears.net/menu/menus.htm and you will probably find that there are more options than you would have thought!
One more note - at CS restaurants, there are a number without veg options on the kids' menu (most CS places seemed to have 3 options on the kids' menu). I asked for a grilled cheese sandwich at most places, and they made one (even if it was just cheese melted on a burger bun!).