Provided that the non-U.S. citizen was born after January 1, 1972, and successfully passed Hunter Education training, they can purchase a hunting license. If non-U.S. citizens meet residency requirements, they can buy resident licenses. If they do not meet residency requirements, they need to pay out of state fees. Under RCW 9.41.175, nonimmigrant aliens may possess firearms in the U.S. if they possess a valid hunting license issued by a state. See http://www.atf.gov/firearms/faq/nonimmigrant-aliens.html.
9th of July, 2012