State club: Washington Falconer's Association : http://www.wafalconers.org/

Washington State Department of Fish and Wildlife
600 Capitol Way North
Olympia, WA 98502-1091
(360) 902-2200 (ask for the Falconry Division/Department)

They'll send you a packet containing:

-Potential sponsors in your region. (You can give them a general area of where you will travel to; I said King County and South Pierce County, so they sent me two regions. Note: Even if these people are listed on the Potential Sponsor list, it's no guarantee they'll even take apprentices.)
-Joint State and Federal Falconry Permit Application
-WA state requirements for obtaining a falconry permit
-Guidelines on what your sponsor will expect of you
-WA state Legislation and laws on Falconry
-The Mews/Facilities Inspection requirements and checklist
-Information on NAFA and how to join NAFA (NOT a requirement for a falconry permit!)

At this time, there is no dedicated WA state Falconry website, unfortunately. You just have to call them to get the info, and it's not a very well put together packet either.

Also note, that WA state requires you to have a sponsor to sign your application before they will contact you to schedule you for your exam.