South Carolina taxes the military pay of residents on active duty, but pay you earn in a combat zone that is exempt from federal tax is also exempt from South Carolina tax. You also get an extension to file and pay state taxes until 180 days after you leave the combat zone, or after you are released from hospitalization for injuries you received there.
There is nothing to apply for. Exclude your combat zone pay from the income you report on your South Carolina return. If you need the filing extension, contact the South Carolina Department of Revenue and reference Revenue Procedure #08-3.
Your Leave and Earnings Statements or orders showing the combat zone service dates, and hospital records if you are claiming the extension for injuries.