Mage: I just pulled my panels off of my 70 GTO a few weeks ago and had the same problem. The vinyl is original and pretty much perfect, so I don't want to get into rebuilding a door panel right now.
I actually got some aluminum sheet at Home Depot (22 gauge, I think?), cut some 2"square patches, and drilled a hole in them. Then I slipped them between the vinyl and the cardboard to give the clip something to bite into. Seems to be holding pretty well so far.
Just be careful pulling the vinyl away from the edges of the cardboard. I found it real easy to rip it and the cardboard backer. Use some 3M weatherstrip adhesive and clamp the vinyl back around the panel. Maybe this helps?