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This info is several years old, but I think it is correct. When I was working out of Seattle several years ago, I went to the King County Courthouse and applied for a permit, waited a little over an hour for them to check the info and walked out with the permit. I did have a Florida permit, but don't remember if that was a factor in the on the spot issuance of the WA permit-might have been.
They may have changed since then, so check it out. If it is still the same you could just get a permit as soon as you arrive and have it for your vacation.

The way this came about was I asked a policeman at SEATAC airport and he didn't have a clue, so I went down to the courthouse to inquire and the clerk gave me the forms. I had not expected to be able to do it all at that time! The permit at that time did not have a photo, just a paper form, but it did the job. This was in about 1990 (I was a pilot flying out of Seattle)