If you are a military spouse who is not a South Carolina resident, you do not pay South Carolina income tax on wages you earn in South Carolina. You still pay tax, if any is owed, to the state where you are a resident.
There is nothing to apply for with South Carolina. Give your employer your residency information so South Carolina tax is not withheld, and file in your state of residence. If South Carolina tax was withheld anyway, file a South Carolina return to claim the refund.
Your marriage certificate, your spouse’s military orders assigning them to South Carolina, and proof of your shared state of residence such as a driver license or voter registration.