Don't know about RV parks here in the columbia area but I'll offer up the following.
1) Sesquicentennial State Park - Near Fort Jackson. Go here for more details: About This Park - Sesquicentennial State Park - South Carolina State Parks
2) Woodsmoke Campground. I have never stayed there but my mother did in a 38ft mh years ago. It is between Columbia and White Rock SC which is very close to Lake Murray. Go here: Welcome - Woodsmoke Family Campground
3) I know Ft Jackson has a RV campground. I do not know the requirements. Go here: Weston Lake Recreation Area and RV Park
Good luck and welcome to the area (if all works out)
P.S. Unless you are specifically looking for Lake Murray ( I lived there for a number of years) You might also look at Lake Wateree. Less congested. Between Columbia and Charlotte.