Lambert's Cove has been voted most romantic restaurant on the Vineyard and the food is pretty top notch. My husband and I had dinner at Atria last weekend and were incredibly impressed with the food. I would ask for a table on the outside porch which is a little more romantic than the dining room. I find they have some of the best food on the island. http://www.atriamv.com/
Special occasion plus romantic? I'd go for The Outermost Inn in Aquinnah, The Sweet Life in Oak Bluffs (where Barack took Michelle last summer), Lambert's Cove Inn in West Tisbury or L'etoile in Edgartown. I have had romantic interludes at all of these and many others- but these are my favorites. And try going in the early fall if you can. You'll have a much more romantic experience.
Sweet Life Cafe
63 Circuit Ave, Vineyard Haven, MA 02568
22 N Water St, Edgartown, MA 02539
Lambert's Cove Inn & Restaurant
90 Manaquayak Road, West Tisbury, MA 02575
Probably best to post to the Southern New England Board as you are apt to get more responses. Also, let people know what part of the Vineyard you will be staying in. Many of the restaurants I've visited are more casual. Take a look at the Mediterrean in Oak Bluffs. I went to their old location in Vineyard Haven and the food was tremendous. They've sinced moved into the old Lola's location which is much larger and I've heard good things. You may also want to look at http://www.mvol.com The Vineyard is a wonderful spot, have fun.