The Horry County Clerk of Court office will process your passport application.
From the Clerk of Courts website:
The processing of a new passport application has to be done in person. You need to bring with you your birth certificate, one 2x2 photo, and a pictured ID (copy of front and back of ID is required by the State Department). The application must be signed in the presence of a clerk. The fee is $110 for adults and $80 for children under 16. This must be in the form of a check, money order, or cashiers check payable to the US Department of State. In addition, the Clerk of Court charges a $25 fee which is payable in cash or money order made payable to the Horry County Clerk of Court.
Clerk of Court - Horry County Government
You may have to call and make an appointment 915-5080.