Virtual photons do not pass from one plate to the other. The leave a plate and then return back to the plate they left from - all in a very short time. The other misconception is that the non conducting plate will block the virtual photons. It may be true that the plate would block real photons of certain energies, but virtual photons are not real photons - they can pass through non conducting bodies. On the other hand, the virtual photons will interract with the plate and cause the atoms of the plate to become small electric dipoles. These dipoles will become additional sources of virtual photons that will will change the electric field between the plates. This is the way dielectrics in capacitors change the capacitance.
The whole concept of virtual photons are really just expressions of how we are able to calculate how a test charge will interract with a electromagnetic field by using Feynman diagrams. It is not a good idea to read too much "reality" into the presence of virtual particles. The results of calculations of the Feynman diagrams is the really meaningful physical result, virtual photons are kind of a hand wavy tool to explain Feynman diagrams.