I'm not the promised expert but since he hasn't made it yet I thought I'd put my two cents in.
I'm hoping you are planning this for next year versus next month since it's going to be hard to find what you want on short notice.
There is a website
http://www.vrbo.com/vacation-rentals/usa/washington/san-juan-islandswhich you might want to look at. It's apparently a list of privately owned vacation homes for rent.
I don't see any large rentals on San Juan island but on Orcas there is one complex of 3 cabins that might work for you:
http://www.vrbo.com/60952 You would have to contact them directly to see if your group would really fit. I'd also contact the owners of some of the larger units and ask them if they know of another large unit nearby.
Bayside Cottages
http://www.orcas1.com/ has 9 cabins - some of them quite large.
Orcas Hotel:
http://www.orcashotel.com/rooms.html 12 rooms.
You might also try the Inn at Ship Bay. They have 11 rooms (not sure of the occupancy total) and I know they sometimes rent all 11 to groups. Might be pricey tho.
On Lopez there is one large rental
http://www.vrbo.com/173041 which talks about having another cabin. Can't tell about the size of the second unit. Again ask them about adjacent units.
Other large vacation rental companies are:
http://www.windermerevacationrentals.com/http://www.sanjuanpm.com/http://hutz.com/rentals