The San Juan Islands have a ton of bed and breakfasts, which I found out when looking for one last summer/fall with my girlfriend. We ended up picking
this one and it was a good time. The breakfast part of "bed and breakfast" was great, probably the best I've experienced, and the rooms were nice enough. It doesn't say if they allow dogs though; we didn't have one ourselves. I skimmed a couple other properties and didn't see any mention of dogs/pets anywhere, so you might need to call in and find out.