On both passenger and driver doors, there is a large central panel where vinyl is somewhat peeling from its base which appears to be some form of dense foam. It's hard to tell, but I'm sure the vinyl should be attached to the foam with some form of adhesive. I don't want to use the wrong type of adhesive which might compromise the foam. Any suggestions?

John
Near Baltimore

You might want to check with a local long-time-in-business auto interiors & tops shop. Quite a few folks (including me) have used the proper 3M spray-on adhesive for applications such as yours....

Many fabric shops also have that and other suitable adhesives.