If you are the nonresident spouse of a nonresident service member who is stationed in Missouri under military orders, the wages you earn in Missouri are exempt from Missouri income tax. You still owe tax, if any is due, to your state of residence. For income tax purposes you may choose your service member's state of residence, your own state of residence, or your service member's permanent duty station.
There is nothing to apply for. Do not report your Missouri wages as Missouri income, and ask your employer to stop Missouri withholding. If Missouri tax was already withheld, file a Missouri return to claim the refund.
Your marriage certificate, your service member's orders assigning them to Missouri, and proof of the state of residence you are claiming.