While vegetarians can eat junk food like everyone else (french fries are typically perfectly vegan

), generally we aim for healthier choices. As a result, we've often turned away from quick service, because it is definitely a lot harder to keep to healthier choices than it would be in a full-service restaurant. It seems that even the things that look like they might be healthy are laden with sugary and/or fatty sauces, or rely on unhealthy condiments like mayonnaise to make them palatable.
At the Magic Kingdom, we used to head to Cosmic Rays, but even there the pickings are slim. We've aimed for Columbia Harbor House on recent trips, and that's worked out okay. We had the Lighthouse Sandwich last time, I believe. Avoid the quick service eatery in Fantasyland.
At Epcot, Sunshine Seasons offers such a vast array of choices that there is always something decent available. The challenge is avoiding all the unhealthy choices, many of which look so much better.

Avoid Electric Umbrella.
At Hollywood Studios, we've had the most luck at the ABC Commissary.
At Animal Kingdom, Tusker House used to be our go-to quick service eatery... of course, it was so good that now it's no longer quick service.

Truly, now, there's nothing, so we still go to Tusker House.