I just thought of one other possibility... The newspaper on the Island is the Beach Comber, their website is: http://www.vashonbeachcomber.com/ They currently have 3 rentals that allow pets depending on the size... maybe if you offered a larger pet deposit they'd consider a larger dog? The other option is to place an add yourself.
The Vashon Island Coffee Roasterie has does not have a bulletin board, but they do have the side of a refrigerator where people are allowed to post items. There address is 19529 Vashon Hwy SW and their phone number is: 206-463-9800. They are VERY nice people and may even post one for you over the phone, who knows? Good Luck! Betty