WA DOL is currently refusing on no grounds except that the FBI will not do background checks for non-LEO reasons - so they are "unable" to process. AFL's are a "may" issue, not "shall" issue, so it is a slightly different matter legally compared to carry permits, but you can't get a carry permit as a legal permanent resident alien unless you have a AFL.
From WA DOL website:
"We are unable to issue alien firearms licenses at this time. The Federal Bureau of Investigation (FBI) has told law enforcement agencies it is against federal law to use federal databases for background checks if the law enforcement agencies share the results with a non-criminal justice agency such as the Department of Licensing. This means law enforcement agencies cannot perform the background checks required by state law for issuing an alien firearms license. Without these background checks, we cannot complete the application process or issue alien firearms licenses. For more information, please call us at (360) 664-6616 or email us at
[email protected]."
Alien Firearms License requirements are in
RCW 9.41.170