No, they do not.
According to the South Carolina Department of Natural Resources, “any vessels that are documented by the United States Coast Guard can be operated on the waters of this state with no additional S. C. registration issued by South Carolina Department of Resources. Documented vessels are exempt from registration in South Carolina.”
However, “if the documented vessel is propelled by outboard motor(s), the outboard motor is required to be titled if the state of principal use is South Carolina, and it is five (5) horsepower or greater. The motor(s) will be titled based on the purchasing situation.”