I have to second what Mischief_Managed said about the veggie sandwich at The Plaza. It's the best sandwich!!!
I third this sandwich as well...very delicious. Sometimes one of their soups of the day is vegetarian chili as well, although I haven't tried it.
I also really liked the lighthouse sandwich and vegetarian chili at Columbia Harbour House at MK. When it's open, the vegetarian tacos at El Pirata Y El Perico are tasty. Unfortunately this is closed for most of the year. If you're going to opt for veggie burgers in the MK, I'd recommend Cosmic Rays over Pecos Bill. Maybe I just had a bad experience, but the one time I got a veggie burger at Pecos Bill, it was as if the bun and burger had been frozen together and then microwaved so that the bun was both soggy and dried out and stuck to the veggie burger patty and it was just all around not very appetizing.
Another excellent vegetarian sandwich is on the lunch menu at the Wave at the Contemporary...so good.
I've heard good things about the grilled vegetable sandwich at the Backlot Express at Hollywood Studios, but haven't tried it personally as it has mushrooms and sundried tomato pesto which I do not like.
At DHS for CS, I usually stick to Pizza Planet cheese pizza. Not amazing pizza, but it's decent and there's ranch dressing packets from the salads they sell and dipping a pizza that's only "OK" in ranch dressing improves it a lot. I think my favorite TS meal at Hollywood Studios is at Brown Derby...the wok-fried coconut tofu noodle bowl (without mushrooms). I've had a good flatbread pizza at Mama Melrose as well. Lots of people really love the stuffed pepper at 50's Primetime, but it's a little too tomato-y for me (I'm picky...I admit it). I haven't tried the tofu dish at Sci Fi, but that's an option as well.
At Epcot, for CS, Sunshine Seasons, and for TS, I'd have to say the vegetarian combination plate at San Angel Inn or the vegetarian option at Teppan Edo. At one time, Le Cellier had this delicious stuffed pepper dish, but the menu has changed many times since then and last time I ate there a mushroom-based dish was the sole veggie option, which again, I personally do not care for.
At Animal Kingdom, for CS I used to go to Tusker House, but since that's now TS, I haven't tried it, so Pizzafari is usually where I end up. The fruit plate at Flametree is good for what it is, but I don't really feel like it's a meal..as for TS I would *NOT* recommend for vegetarians here, Yak and Yeti...they had pretty much nothing on the menu vegetarian - I ended up having to eat a salad with the meat left off. I've heard that now that Tusker House is TS, it does have decent veggie options, so it's probably one to try.
Boma at Animal Kingdom Lodge has always been a favorite in my family, and the servers or the chef are always more than happy to point out what on the buffet is vegetarian.