When I read your question I realizes that even though I've lived in Washington all my life,I didn't know the answer!
So I followed Gary's link and it looks like you can get a license without a street address,but have to apply in person.
http://www.dol.wa.gov/driverslicense/moving.html
Moving? Get WA license
By law, you have 30 days to get your Washington driver license once you establish residency in Washington State.
How to establish residency in Washington
You?re considered a Washington resident when you do any of the following:
Arrive and begin establishing a home in Washington.
Register to vote.
Receive state benefits.
Get any Washington State license at resident rates.
Receive in-state tuition fees.
How to get a Washington State driver license
Find out if you can pre-apply online.
If you?re a Washington resident, 18 or older, have never had a Washington driver license, instruction permit, or ID card, and have a Washington residence street address, Pre-apply online and save time in the office. You?ll finish your application at a driver licensing office.
If you,
Don?t have a Washington residence street address, have other special address issues, or currently have or in the past you?ve had a Washington driver license, instruction permit, or ID card. Apply at a driver licensing office.
Are upgrading your license or ID card to an EDL or EID. Apply at a EDL/EID office.
Visit a driver licensing office to start your application or to finish an application you started online. Bring the following with you:
Proof of identity, including your out-of-state license.
Payment for the fees.
Your Social Security number (mandatory to help enforce child support laws).
You don?t need to show us your card, we?ll still be able to verify it.
If you don?t have a Social Security number, you must verify your Washington residence address before you can proceed.
Pass a vision screening.
Pass any required tests:
If you have? Pass these tests?
A valid license from:
Another U.S. state
British Columbia
Germany
South Korea
Taiwan (Taiwan Driver?s License holders wanting to obtain a Washington State Driver?s License, please contact the Taipei Economic and Cultural Office in Seattle (TECO) at 206.441.4586 ext. 311 or by e-mail at
[email protected])
No tests required.
A valid license, but have a medical or physical condition that shows testing is required Knowledge and drive test
An expired license from another state Knowledge and drive test