There appears to be a cure for vinyl shrinkage and adhesive failure with these things. It's called do-it-yourself leather.
First, we used the techniques listed
here to clean & re-create the squishy layer between the outer cover, and the storage box's door. The only part of the original vinyl cover which was kept, was the area directly behind the latch handle; polyurethane adhesive was used to glue the vinyl to the door.
(the silver shiney stuff is run-of-the-mill carpet undermat, about ¼" thick)