Thank you for your enquiry. You can only park a campervan at the Giant's Causeway Visitor Centre during the day, it is not possible to stay there overnight. There is another carpark nearby but it also shuts at nighttime. There are a couple of caravan/campervan parks located a few miles from the Giant's Causeway. One is Ballyness Caravan Park which has been awarded a 5* rating and the other is Bush Caravan Park which has a 4* rating. Both are in the vicinity of the town of Bushmills.
Another option for you might be to stay in the nearby seaside town of Portrush, as there are a few caravan/campervan parks there. Details for all the sites can be found on our web site: www.visitcausewaycoastandglens.com
Portrush is on the bus route of the Causeway Rambler Bus (402). The Rambler bus is a hop-on, hop-off service and it stops at many of the local attractions such as Dunluce Castle, Giant's Causeway, Carrick-A-Rede Rope Bridge and the Bushmills Whiskey Distillery. You can find the timetable for it at: www.translink.co.uk
I hope this gives you a few options for staying in your campervan when you visit the Giant's Causeway.