That's correct. You can successfully complete an EMT-B course in Washington State, but if you are not affiliated with an EMS organization, you will NOT be granted a Washington State EMT certificate. You can still take the class you will have to take the National Registry, but that really only comes in handy if you plan to move to another state.
Affiliation comes from a job with an Ambulance Company or Fire Department or a volunteer position with Fire, Rescue, and in some cases, Sheriff's Office Search and Rescue. Also worth noting, it may take upwards of 6 weeks to obtain your cert once you are granted affiliation by an organization.
Where in Washington are you? Hopefully not Western Washington, as the outlook for employment for potential EMT-Bs here is dismal, at best.