Your vet will give you the health certs for moving once you let him know the date.
Your vet is also the best person to tell you what shots he recommends for the AREA of SC you are moving to.
If he's not sure, he will call a vet in that area or at least he better, since that's how my vet did when I moved my horses from PA to SoCal, and likewise the SoCal vet did when I moved them from SoCal to TN.
I am of no help regarding shippers, because I trucked my horses myself.
Good luck and good fortune in your move! 