Hello,
I have a few favorites. In MK, I like the Columbia Harbor House, they have two veggie options: veggie chili and the Lighthouse sandwich. I have never had the chili, but the sandwich is essentially hummus and broccoli slaw on toasted bread. Pecos Bill also serves veggie burgers and they cook them away from the regular burgers to avoid cross-contamination.
At Epcot, I typically go the Tangierine Cafe and get the dish that is pictured above. Sunshine Seasons also has some veggie options, but they are as good as they used to be, IMO. The last time I ate there I got chik'n and stir fry noodles. There were no vegetables. Hello?! We're VEGetarians, we like the vegetables.
In all the years I've been going to WDW, I've never eaten at the Studios. I find DAK lacking for veggie options, but I know they have a QS place that has a veggie noodle salad and veggie spring/summer rolls.
I like the Flying Fish Cafe. It's not always on the menu, but you can ask your server for the "bento box." It is served on a dish with four compartments and each compartment has a side dish of veggies and one has their awesome risotto. The servers have always asked me my definition of vegetarian, so there is no ambiguity about what to give me.
The Captan's Grill is good for breakfast. I like the buffet, because it has a lot of variety for veggies.
Jiko, Boma and Sanaa also have lots of great options for vegetarians. It helps to like Indian food, though!
This was off the top of my head. I'll see what else I can think and post them.