"The membrane is made of polyethylene (PE) and the case is made of polyvinyl chloride (PVC
is only a problem when it contains plasticizers. Hard PVC
, used for window frames, is a very stable material.
Pure PE is also unproblematic as long as you keep it out of direct sunlight. Nevertheless it could contain small amounts of stabilizers etc..
But it is very unlikely that the foil of the coin display is pure PE. PE is a semicrystalline polymer and therfore not completely transparent. You know the material from plastic bags. Also I don't think that a PE foil would be able to keep a coin in place, it's not sticky enough.
I rather think that the foil is a PE-copolymer, Poly(ethylene-butylene), or something similar. Also this material is unproblematic if it is used without any additives, which is possible, and maybe "archival safe material" means that it is without additives. Unfortunatelly there are plasticizers for these copolymers, and as long as I do not know for sure if these foils contain plasticizers or not I would keep my fingers off, but maybe I am overcautious.