Magic Kingdom:
Columbia Harbor House - Lighthouse Sandwich, vegetarian chilli (real delish)
Tortuga Tavern - vegetarian burrito BUT you can get the burrito contents in a taco salad bowl and its way better
Pecos Bills/Cosmic Rays - Veggie burger
(I reccomend the first two options!)
Epcot:
Sunshine Seasons - tons of options! tofu and veg noodle bowl, sushi, tabouleh pinwheels, veggie ruben etc...
Morocco - my fav! vegetarian platter - omit the lentil salad for vegan/dairy aversion
Hollywood Studios:
Backlot Express - grilled vegetable sandwich (I haven't tried this one - we usually do 50s/Sci Fi while at this park for lunch..)
Pizza Planet - I haven't tried this one - I don't do dairy, but the atmosphere is pretty cool
Animal Kingdom:
Restaurantasaurus - vegetable subosaurus with sweet potato fries (make sure you snag a seat inside the airstream trailer, and read the stories around the restaurant!) This one is OK, not amazing, but its a decent option if you don't want to leave the park
Resorts:
Art of Animation - amazing veg options! home made veggie burger - 99% sure its a chickpea patty with edamame and other veggie, it's real good!Tandoori aacorn squash, smoothies for drink/dessert and best of all babycakes mini cupcakes for dessert or snack!
Grand Floridian - tabbouleh wrap, real yum and filling, also a visit to that resort is a great break from the parks
Animal Kingdom Lodge - at the Mara I believe they do falafel, and I do believe it was great last time I had it
Roaring Forks - Veggie sandwich with hummus and roasted red peppers, really tasty
Downtown Dis:
Wolfgang Puck Express - I prefer the one in the Marketplace, it's a little newer and they recently expanded their indoor seating area. Lots of veg options from fresh thin crust pizzas and pastas, and if he's just a vegetarian you can get creme brulee as a dessert with a QS credit.
Definitely don't miss Morocco, or Columbia Harbor House. Also, if you're at Columbia - go upstairs and sit on the overpass where you can people watch in front of the Haunted Mansion.