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There are some great vegetarian selections at WDW. A few of my favorites:
Columbia Harbour House in MK: The Lighthouse sandwich and the vegetarian chili are both very good.
Liberty Tree Tavern in MK: The Amber Waves of Grain (available at lunch) is delicious.
Plaza Restaurant in MK: Wrapped Up in the Garden was very good. Looks like they recently switched to a vegetarian sandwich on foccaccia bread, but the ingredients look the same as what was in the wrap.
Sunshine Seasons at Epcot: Several great selections, but I really liked the roasted beets and goat cheese salad and the grilled vegetable Cuban sandwich.
Tangerine Cafe: The vegetarian platter is very good!
Cantina de San Angel: There aren't any vegetarian selections listed on the menu, but you can ask for pretty much anything here without meat, and they will fix it for you. I've had nachos and burritos without meat, and both were very good.
Chefs de France: The vegetarian selection here changes often, but everything we've tried has been delicious.
Toy Story Pizza Planet at MGM: The veggie pizza was really good, and you can get a combination that includes a side salad.
Earl of Sandwich at DTD: The Caprese sandwich is wonderful!
Rainforest Cafe at DTD: I think they have the best veggie burger on WDW property.
Yacht Club Galley: The tomato and mozzarella sandwich at lunch is delicious.
ESPN Club on the Boardwalk: The grilled portobello sandwich is great.