What parks will you be in?
I'm vegetarian (lacto-ovo--I eat dairy and eggs). Most table service restaurants have one veggie entree. Most counter service kiosks do, too. Some of my favorite places to eat in the parks are:
Magic Kingdom--I like the Lighthouse Sandwich at Columbia Harbor House. Main Street Bakery usually has a vegetarian sandwich that's very tasty, but it's often not sitting out (you have to ask for it).Both of these are counter service.
Animal Kingdom--not much to eat for us vegetarians. I usually have pizza at Pizzafari (the decor is fantastic, by the way!

). I LOVED the pizza and the veggie burger at Rainforest Cafe.
Epcot--my favorite table service meals are the macaroni and cheese at Chefs de France, the vegetarian plate at San Angel Inn in Mexico (ask for the vegetarian menu--it's not on the regular menu), and vegetable curry at the Rose and Crown. My favorite counter service at Epcot is Sunshine Seasons for the grilled vegetable cuban sandwich. I also like to get the chips at the fish and chips kiosk in the UK--they are fried separately from the fish so they are tryuly vegetarian.
Studios--We don't spend a lot of time at the Studios, but ate at Mama Melrose earlier this year and loved it! Hated the food at the ABC comissary (nasty vegetarain wrap that made me queasy).
Be careful at the counter services in the World Showcase where the cast members don't speak English well--at the Mexican Cantina, the cast members clearly did not understand what I was asking re: vegetarian dishes. They used to have a vegetarian black bean burrito dish but the last time I was there they only offered a salad for vegetarians.
Your friend will find plenty of food as long as she is not vegan--there don't seem to be a lot of vegan entres.