I'm not sure about ordering vegan off the menu, but at any of the finer dining restaurants you can request meals with special dietary restrictions. We have dealt MANY times with this for a daughter with an odd food allergy and have had amazing meals prepared for her.
Our favorite places are Storytellers Cafe, Napa Rose, Blue Bayou and (for breakfast) Rainforest Cafe. We have had the most success when we had reservations and called ahead to let them know her restrictions.