I don't think there are too many directories of children's institutions for the State of South Carolina, but try this one :
South Carolina Legislative Manual.
It might have been published annually.
The problem with orphanage records is not what is the law? The problem is that orphanages no longer exist. Foster family homes have replaced them, except where the child has a problem of some sort.
Carolina Children's Home had a mailing address in 1977: Box 4465, Columbia, SC 29240. It must have kept a low profile because it is missing from some directories. It was picked up in the special census of 1933, which lists no address for it. As far as I know this census only collected statistics, not names of the children. 104 dependent white children were in Carolina Orphan Home in 1933.
Sometimes orphanage records are acquired by historical societies (in the state capital or the county seat) and libraries. Now and then someone creates a web site devoted to an individual institution, and a Google search might find it.
The inhabitants of orphanages are listed in the federal censuses, such as 1930. The 1940 census will be made public 72 years after it was taken.