You cannot buy online and have a firearm shipped to you unless you hold an FFL from the BATFE. Since you're not a dealer and are not a collector, I'm going to assume that you do not have an FFL.
I am also in Washington State, here are the laws:
You will need to go, in person, to a firearms dealer to buy. If you find a firearm you like online, the dealer may (and I stress "may") allow you to have the firearm shipped to the dealer (for a fee that ranges from $30 to $50 depending on the dealer), who will then complete the transfer as required by law.
Since you presumably do not have a Washington state CPL, you will fill out all applicable paperwork, pay for the firearm, and then leave with a receipt (but not the firearm).
After the WA state background check comes through and the waiting period has passed, the dealer will call you and you can then return to pick up the firearm.
Be aware that many gun shops will refuse to transfer firearms that they can order themselves, so unless you're buying a used firearm online, you may have some difficulty finding a dealer who will transfer the firearm for you.
EDIT: Wade's in Bellevue is a good shop. Their prices are higher than some, but not bad and they have a very nice indoor range with numerous firearms available for range rental. It's a good setup to allow you to try out different options.
ZV