If you are a nonresident military spouse stationed in Vermont with your service member, the wages you earn in Vermont can be exempt from Vermont income tax.
There is nothing to file with the state in advance. Claim the exemption on your Vermont return, and ask your employer to stop withholding Vermont tax from your wages.
Your marriage certificate, your service member’s military orders assigning them to Vermont, and proof of the state you were both residents of before the orders.