I take it your swing is on a frame? If so the vine will grow much like a morning glory, at least it does for me. It just needs something to vine around and it will grow up.

Be careful, Moonvine will grown on anything and will overgrow most things from my experience with them. I had some last year that reeked havok on a metal arch. Had a treated post 4X4 arch concreted into the ground for them this year. They are worth the effort though.