Can a person precluded by law from possessing a firearm hunt in Washington with archery equipment when legal to do so?

Yes. Such individuals can purchase an archery deer or elk tag or modern firearm deer or elk tag and legally hunt during those seasons using only legal archery equipment.
Such individuals could also hunt with a crossbow during the modern firearm season in firearm restricted areas if they purchased a modern firearm deer and/or elk tag.
They could apply for special permits for bear, cougar, goat, sheep, or moose and if drawn hunt only with legal archery equipment.