Unlike in Washington, there is no separate restoration procedure under federal law.
Therefore, the dispositive question is whether your federal firearm ban is based upon a Washington state conviction.
Does Washington's restoration procedure allow you to possess firearms under federal law?
Yes ... if you have a Washington State conviction
Under 18 U.S.C. 921(a)(20), whether you have a "conviction" for federal purposes of a federal gun ban is determined by "the law of the jurisdiction in which the proceedings were held."
That is, if your federal firearm ban is based upon a Washington State conviction, then you can lift your felony ban through Washington's restoration procedure in state court.
No ... if you have a federal conviction
The U.S. Supreme Court has ruled that a state restoration procedure does not remove a federal firearm ban if the ban arose from a federal felony conviction (i.e. a conviction in federal court).
Your only option is to apply for a presidential pardon, which is rarely if ever granted.