I've been a vegetarian for about 20 years now, and I think you really do need to plan carefully to find things to eat. True, most places have something veggie on the menu, but you could well be stuck with cheese pizza or a side salad if you're not careful.
There ARE lots of veg-friendly WDW restaurants - make your ADR's so you're not stuck. We always love Boma, the Crystal Palace buffet, California Grill (it's not on the menu, but they will make an awesome vegetarian plate), and Tusker House.
Any of the nicer restaurants are going to be accomadating, but it is luck of the draw with the chef - are they creative enough to come up with an interesting off-the-menu veggie dish, or are they just going to grab some pasta and cut up some zucchini and hope you're happy?
One last thought - if you go during F + W festival, I have always found plenty of veggie stuff from the carts.