Multnomah county does as well.
It requires a safety class (no shooting requirement), background check, letter from WA state dept of mental health stating you have not been involuntarily comitted for a erpiod greater than two weeks, and a letter to the country sheriff's office stating why you want it (which can be "sporting use" or "self defense".) Not much detail is required.
Multnomah county only accepts the application in one place, and they are super strict about making your appointment on time. It's a pain if you have to cross city traffic.
Mine was turned down for having missed a court date for a traffic issuewithin the last 4 years. I intend to apply again in April when that no longer applies. I will have to re-take the class (has to be within 6 months of application) and jump through all the hoops again.
Rumour has it that they are pretty picky about people who don't live in border counties. The fee is of course, non-refundable.