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Disneyland web site says you can call ahead regarding dietary restrictions, but it's very vague about what can be done for vegetarians/vegans. I've never tried this myself. I am also vegan, and generally cannot find much to eat other than fruit or plain salad (which I don't much care for). WDW generally is much more vegan/vegetarian friendly.
Some items I've run across at DL:
The vegetarian gumbo at Royal Street Veranda in New Orleans Square is supposed to be vegan. However, the bread bowl it comes in is not vegan. You can request a plastic bowl or large paper cup instead of a the bread bowl, though. This obviously would not be as filling as having the bread bowl with it.
French Market or Cafe Orleans in New Orleans Square (I forget which) has a vegetable stew in a non-vegan bread bowl, but I'm not sure if the stew is vegan.
Hungry Bear Restaurant in Critter Country is supposed to have a vegetarian burger, but I'm not sure if that's vegan, either. I suspect it probably isn't.
The vegetable skewers at Bengal Barbeque in Adventureland is basically vegan, but it is cooked along with the meat skewers, so DL doesn't call it "vegetarian."
Rancho del Zocolo in Frontierland can make a burrito or other items minus the meat and cheese, and it will supposedly be vegetarian/vegan.
Redd Rocket's Pizza Port in Tomorrowland has eggless pasta (or it did when I checked a couple of years ago). However, they toss the pasta in cheese. You can request that they make a fresh batch for you without tossing it in cheese then top it with plain marinara sauce. This takes a while, though, and it winds up tasting like pasta with a can of tomato paste dumped on it.
As you can see, the pickings are pretty slim. When I visit DL, I generally wind up packing my own food or eating elsewhere. DCA is even worse. There's basically nothing except fruit. You might be able to find food at Downtown Disney restaurants, but you'll have to ask lots of questions and/or request that they alter dishes on their menus -- pasta minus the meat and cheese, for example. This would probably be much more satisfying than trying to find tasty and filling vegan food in the parks themselves. There are a few vegetarian places outside of DL around Orange County, but of course this is not as convenient as eating on property.