These are all Bainbridge Island places!
Bainbridge Bakers -- they have an outside area, dog biscuits, dog bowl and they're right next to a park. They don't mind if you hold your dog while you order, either. NOTE: Do not get this place confused with Blackbird Bakery which is on the way up if you're walking from the ferry - they do not allow dogs inside (found that out the hard way).
Streamliner Diner -- outside seating area allows dogs, you won't have to go in to order because a waiter/waitress will come up to you.
A Some Pizza -- Outside eating area allows dogs
The Lookout on Capitol Hill (Will is often a co-host at trivia on Tuesdays)
Nollie's Bakery in SLU