Article gets into this a bit: "Eric Goldman, a professor at Santa Clara University School of Law, said the First Amendment and Section 230, a bill that shields internet providers and tech companies from liability for the posts, photos and videos people share on their sites, provide a strong defense in many instances for websites and providers facing lawsuits over hosting information about abortion access."
While none of those bills related to abortion, I'm still skeptical the courts would let this bill stand
It's more about scoring brownie points with your base and tying up your opponents in litigation for years... Even if they know the law is blatantly unconstitutional, there's no real downside to the legislatures (assuming their constituents support the unconstitutional law)
Where this would fall afoul is likely interstate commerce. South Carolina can't declare a website hosted in California illegal due to jurisdiction. They might be able to convince ISPs to block them, though.