There are a lot of options to get your veggies at.
MK - Columbia Harbor house - you can get a broccoli peppercorn salad with or without chicken they also have sides of steamed veggie and couscous , Cosmic Rays - Greek salad with chicken, I think you can request it without the chicken, also we like the liberty square market, lots of choice for fruit and veggies here, and they have roasted corn on the cob at times.
Epcot - Sunshine Season would be the go to for this. lots to choose from and the menu changes up, We haven't eaten here in a while so not sure. Walking around the world showcase there are a ton of options
Animal Kingdom - Flame Tree - Mixed green salad with roasted chicken, I think it called Kamal's it's a snack cart and it offer hummus and veggies.
Hollywood Studios - ABC commissary come to mind - They have a Mediterranean salad, with or with out chicken
Another thought, if you leave the parks you open up a whole a lot of choices... If your in Epcot head out the back gate to one of the resort Food Courts on the boardwalk , don't forget about the Swan and Dolphin, At MK just a monorail ride away to the resort on the loop, Hollywood studios take a boat over to the resort on the boardwalk. AK- you could head over to AKL and check out what they have to offer,
Some of the resort offer make your own type salad and pasta offering...