You do not pay Oregon income tax on wages you earn in Oregon if you are a nonresident military spouse. You still have to file an income tax return in your own state or territory of residence if that state requires one.
There is nothing to apply for. Leave your Oregon wages off an Oregon return, and give your Oregon employer your residency information so they stop withholding Oregon tax. If Oregon tax was already withheld, file an Oregon return to claim the refund.
Your marriage certificate, your service member spouse’s orders assigning them to Oregon, and proof that you both remain residents of another state or territory.