If you are the nonresident spouse of a nonresident service member stationed in Ohio under military orders, you do not pay Ohio income tax or Ohio school district tax. For income tax purposes you can choose the service member’s state of residence, your own state of residence, or the service member’s permanent duty station as your state of residence.
There is nothing to apply for in advance. File as a nonresident and, if Ohio tax was withheld from your wages, file an Ohio return to claim the refund. Give your employer updated withholding paperwork so it stops.
Your marriage certificate, the service member’s orders assigning them to Ohio, and proof of the state of residence you are claiming.