Reader Query: Martha’s Vineyard Eats?

Knowledgeable readers, we have a question from reader Brandy H. that I cannot answer. Please help her in the comments!

We are going to Martha’s Vineyard in August (from Los Angeles), where we have rented a house so that I can cook. I am wondering if you have any culinary knowledge of the island? Do the island summer markets feature a unique local specialty? Is the island known for having a particularly heavy crop of strawberries, squash or tomatoes? Is there perhaps an island spin on the ubiquitous lobster roll, clam bake or crab boil? Also, what are the restaurants that should not be missed? In short, I am wondering what should be on the Amateur Gourmet’s dining itinerary for both cooking in, and eating out (fine and casual) on Martha’s Vineyard?