I have to admit that pet friendly accommodations in the San Juan's are not exactly my area of expertise. However I was able to do some checking around and found the following:
Pet Friendly lodging on San Juan Island:www.harrisonhousesuites.com www.tuckerhouse.com Pet friendly accommodations on Orcas Island:The Deer Harbor Inn and restaurant http://www.onceinabluemoonfarm.com/http://www.orcasblueheron.com/http://wwwdoebay.com Cabins, hostels, yurts, and camping $20 pet fee per pet, and that not all spaces allow pets.
Visitors to Orcas Island and the village of Eastsound can enjoy the
Orcas Island Dog park . Friday harbor, on San Juan Island, also has its
very own dog park just outside of town.

Visitors to Friday harbor might also find this
animal daycare service helpful. Another option on San Juan Island is
The Animal inn which offers animal lodging for dogs, cats, birds, plus provides veterinary services, special care needs, playtime, and grooming.
I hope that helps; enjoy your stay in the San Juan's!