If you want a soup but don't want the waste of the breadbowl...you can get the chili/gumbo/soup in a little to-go thing (it looks tiny) and they'll put the bread into a baggie for you...that way you can choose how much you wish to eat, instead of having it decided for you by the bread being "ruined" by having the soup in it.
If it says "vegetarian" it is. The beans and rice at the Mexican restaurants are made with vegetable oil of some sort (NOT lard).
Plaza Inn has a lovely and simple penne pasta with marinara that we love.
You can order things without meat...we get the breakfast burrito at WhiteWater Snacks without the meat (just make sure before taking a bite, or if you ordered to go, before you leave), we get the breakfasts at Tomorrowland Terrace without the meat, etc.
The veggie burgers that they serve are not the soy-type that feel/taste like meat. They are more "vegetable burgers", where you can see the small chunks of carrot and broccoli, etc. I have also gotten black bean burgers before, which actually were NOT appreciated (that's not the taste I'm going for, generally, when I order a veggie burger) but most recently it's been the vegetable burgers.
Check out the menus on wdwinfo.com and especially allears.net.
Cafe Orleans (not counter service but still) has a veg ratatouille, and it was just confirmed that they'll make their dinner crepes vegetarian upon request.
I can't really think of a counter service place (other than the corn dog stands( that won't help a vegetarian...heck, even the hot dog place in the Hollywood section of DCA has a portobello mushroom dog!