My Boyfriend and I have decided to move in together but he currently lives in Ohio and is on Probation for a DUI (second offence) and I live in South Carolina. Can He move to SC while he's on probation?

As long as he requests it from his PO and the PO approves it.

https://www.askmehelpdesk.com/criminal-law/how-transfer-probation-115783.html

Here's a handy post explaining how to go about getting the probation transferred. Your boyfriend should not leave Ohio (even to go visit you in SC) until everything is completed and transferred properly.