Getting so the old glue off is the hard part. The vinyl is peeling because the glue failed, and if you replace the vinyl without getting all the old glue off the new will fail too. BTDT. Peel it all off, clean with whatever solvent seems to cut it. Start with milder solvents first like isopropyl alcohol and work you're way up through the nastier stuff till you find one that works. Wear a filter mask. Then sand. From there you can go with new vinyl and contact cement (also nasty) or paint. I'd go with paint, and good quality exterior latex house paint works surprisingly well in this application. IMO a satin finish looks better for the interior than gloss, and latex does this well and a lot less toxic than oil based.