While it isn't Vegan or Vegetarian, and it probably isn't very healthy (aside from those Omegas) Bubba Gump Shrimp House in Gatlinburg, TN is sustainable seafood. It is a good, family friendly atmosphere. The kids even enjoyed fish in their kid's meal. I actually heard, "It tastes like chicken!".
I had a fish sandwich which was a little on the spicy side, (Cajun, I guess) and it was the best fish sandwich I've ever eaten. My significant other had several things on the menu including a giant cookie dessert. Everyone gave it two thumbs up. When we were finished with our meals, the waitress gave us a quiz over the movie, "Forrest Gump". I was surprised that I knew the answers to several questions. We also liked the feature of the license plates to get the staff's attention. Flip it to "Run Forrest Run", and the staff leaves you alone. Flip it to "Stop Forrest Stop", and any server or staff will stop to serve you. It was a fun feature, but we noticed at least one staffer, avoiding our "stop" signal.
These are chain restaurants so they are all over the south. It is a fun, unique dining experience for the family.