I've been reading alot about the types of foam to use to cover interior door panels. Would 3/8 inch carpet padding work if you were not going to put a design in the panel? I plan on using a hardboard backing, gluing foam to it and then covering the panel with vinyl.

No. Carpet padding is too "chunky" and does not have a smooth surface. The uneven surface will show through your material and you will have a nice lumpy interior.