I'm a FFL and Washington law differs from federal law regarding black powder firearms. Black powder firearms are regulated under Washington law and dealers are required to follow state transfer requirements (ATF form 4473 for rifles and 4473 plus report of pistol transfer for pistols). This follows from RCW 9.41.010, under which a black powder firearm is considered a firearm for the purposes of regulating firearms sales and transfers. And, a person prohibited from owning or possessing a firearm in Washington would also be prohibited from owning or possessing a black powder firearm.
However, I believe most internet retailers will ship directly to customers in Washington State. Buying mail order seems to be legal (unless you're legally prohibited from owning/possessing a firearm), because the Washington law regarding transfer of black powder firearms is imposed on the dealer. If you buy a black powder firearm locally, the dealer should fill out the same paperwork as for a modern firearm.
