The June, 2020 issue of The Skyliner Magazine has an article where the author worked on a 1959 Skyliner to smooth out the vinyl on the door panels. I will attempt to attach the scanned article but it may be too large to post (about 1.5 MB). In any case, he went through the following steps (with much more detail than I have listed).

1) Remove the panels from the car, then carefully remove the stainless trim by prying any attaching metal tabs and also remove any staples through the vinyl to the cardboard backing.
2) Very carefully remove the vinyl from the backing board by applying heat (hair dryer) and very slowly pulling the vinyl from the board to keep from damaging vinyl.
3) Lay the cardboard over a 24" X 48" lauan panel (about 5.2 mm or 3/16" thick) and cut out the panel to match the cardboard.
4) Use 3M adhesive spray to attach the vinyl to the lauan backing and carefully smooth out the vinyl as you go (use heat from hair dryer here also)
5) Reinstall stainless trim and reinstall panels.

I can send you the PDF attachment by email because it exceeds the file size allowed. It has photos that show the procedure better than the limited description above.

Regards,
Dave