Yes, WA State issued a license to legal permanent residents for acquiring firearms, the problem is the DOL and not law enforcement is in charge of it, so the FBI will not the DOL access background checks, hence they cannot issue new licenses.
check this out
http://thehighroad.org/showthread.php?t=297951
Bill HB 2955 became effective on June 12, 2008. This bill authorizes the WA DOL direct access to federal criminal backgrounds records. In a nutshell this law certifies the WA DOL as a "Criminal Justice Agency".
Please contact your local Representative, Representative Ross Hunter & Cathy McMorris Rogers and ask them to contact the FBI regarding the issue as at this time the FBI is stalling on allowing the WA State DOL access to the background check.
Or are they still dragging their heels?