Veg in New Orleans
I wouldn't call it a great restaurant, but Thirteen on Frenchmen Street takes good care of the vegetarians. Perhaps for a light meal so that you can eat more somewhere else?
Most tourist friendly or upscale restaurants will have vegetarian options. Many neighborhood places won't.
I've posted on this before so if you search, you might find my previous comments. My husband and I eloped to New Orleans three years ago last week and had our wedding night dinner at GW Fins in the Quarter. I'm a veg and he's not. Although what I had wasn't listed on the menu, they took great care of me and it was one of the best meals I've had. Nothing super fancy but just let the flavors of the veggies shine thru. If you go, DO NOT miss the mashed sweet potatoes!
We also had an incredibly good dinner at Jacques Imo's.