Yes! My ex-boyfriend is vegan and they were able to fully accommodate him at both the Chef's Counter option (highly recommend), and the regular ala carte menu. They have both special plant-based dishes, and specific vegan dishes. These rotate fairly often, so I'd check the Disneyland.com menu or call to get the latest details.
I'd say it's very likely. They know ingredients to everything, and can modify dishes. Doesn't hurt to call ahead to ask though.
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