It's possible that there is some clown-specific law that I've missed, but it's likely referring to a more general ban against masks and related means of concealing your identity:
Police Chief Ken Miller said that his department would begin enforcing a state law that bars anyone older than age 16 from publicly concealing their identity with items such as a mask for non-employment matters.
The applicable law can be found in SC Title 16, Chapter 7, Article 3:
SECTION 16-7-110. Wearing masks and the like.
No person over sixteen years of age shall appear or enter upon any lane, walk, alley, street, road, public way or highway of this State or upon the public property of the State or of any municipality or county in this State while wearing a mask or other device which conceals his identity. Nor shall any such person demand entrance or admission to or enter upon the premises or into the enclosure or house of any other person while wearing a mask or device which conceals his identity. Nor shall any such person, while wearing a mask or device which conceals his identity, participate in any meeting or demonstration upon the private property of another unless he shall have first obtained the written permission of the owner and the occupant of such property.
There are exceptions to the law for occupational reasons, holiday costumes, etc, found in Section 16-7-130 on the linked page.
The discrepancy in ages (16 v 18) is odd, but the two articles from Reuters appear to have used the same source for them (Chief Ken Miller) and were published only a day apart, so I'm assuming that the 18 was an oversight on their part, and got picked up from there in the other linked stories (both of your other links reference Reuters).