I'm also Ovo-Lacto and haven't had any problems at WDW. I've had trips where we ate mainly CS and trips (with
DDP) where we also ate a lot of TS. Neither was ever a problem, though options can be limited at any one place. The fruit stands/carts are great, too - they fruit is nice, not old and mushy.
Couple of thoughts:
The veggie burgers are Boca. Only you know if that's ok with you. I like them, but not every day.
Check out the menus others have suggested as a start.
Don't worry if your group wants to go to a place where the menu doesn't list veggie items. I have never been to a WDW restaurant that wouldn't accommodate vegetarians somehow. At some, they have things like veggie burgers that just aren't on the printed menu and at some, the chef will ask about your specific needs and fix something up.
If you make any reservations, note that there's a vegetarian in the group and then mention it again when you check in. If you're really worried, you can call the restaurant itself ahead of time and speak with one of the chefs.
The Rasta Pasta (minus chicken) at RFC is out. The sauce isn't vegetarian, but it took a whisper from one staff member after another had lied to me to keep me from being sick all evening....... While some people like their veggie burgers, I find them inedible..... sigh.....
Enjoy!!