The range at Plantation up near Bellingham --
http://www.co.whatcom.wa.us/1931/Plantation-Rifle-Range -- requires shooters under 18 be accompanied by a "parent or guardian." Since you have permission from his parent to take him shooting, that should suffice in making you a guardian under those terms. Really, unless there's some reason an RSO or other range worker might ask if you're his parent, most will probably just assume you are and it will be a non-issue.
Some of the shooting pits in Darrington, about an hour (maybe a little less) east of you on Hwy 530, could be an option if you want something less formal. Those should be on the map MadPick mentioned.