Martha's Vineyard restaurant
The 'better' restaurants do a good job with fish - especially local seasonal catch. One suggestion is State Road Restaurant in N. Tisbury - they serve all day http://www.stateroadmv.com/. Also, Sweet Life Cafe in Oak Bluffs has an excellent reputation. It's pricey but worth it in my opinion. http://www.sweetlifemv.com/
At this time of year, it's best to check with restaurants to see if they are still operating.
Sweet Life Cafe
63 Circuit Ave, Vineyard Haven, MA 02568
State Road Restaurant
688 State Rd, Vineyard Haven, MA 02568