Can you stick PVC on to varnished wood?

Yes you can ... BUT ..... the old varnish was cellulose based, modern "varnishes" are polyurethane based and as such are more harder. It also depends on the on the style and quality of the PVC you wish to adhere to it. To be on the safe side, you can sand the polyurethane to help in adhesion, and if a solvent based adhesive could melt the pvc, .... I would try a waterbased contact adhesive. Apply this product to both surfaces, press them together and immediately pull them apart again, leave until tack free (or nearly dried) and then press them back together again for final adhesion.

Firelighter - Australia