Sky Bar Steak & Sushi has some vegetarian options. The ones that I've had are the seaweed salad, the edamame, and the miso soup. All very good. I'm sure there would be a Calif Roll or other vegetarian rolls and vegetable tempura. It's a bit pricey but we go during Happy Hour though even that can add up depending on drinks and how much sushi you consume.
Mosquito Cafe also offers various vegetarian options. I love their grilled mushroom/vegetable sandwich. I'm pretty sure they offer other items plus daily specials. Just look at the chalkboard straigh in front of you when you walk in.
I was thinking I had heard of a vegetarian resto in Galveston recently but I can't place it. Here is a website listing some options:
I can definitely recommend Mosquito Cafe although I haven't had any of the veggie options. Eatcetera is on my list of places to try and Oasis looks good but I can't comment on either. I was going to say that Kroger probably has a salad bar, too.
I'm not vegetarian (nor from the Island) but I eat a lot of meatless meals. When I think vegetarian here in Houston I always think Indian. There don't appear to be any Indian restos on the Island but I can recommend Cuisine of India in Clear Lake, not that far away. There are probably other Indian restos in the bay area that would be suitable, too.
Here in Houston most restos offer at least some vegetarian options and many will make vegetarian dishes on request. I think many of the restauranteurs in Galveston would be sophisticated enough to handle special requests, too.