Your question mixes together light/matter interactions and light/light interactions. The pressure you refer to is light/matter interactions. Sound propagation in your context would refer to light/light interactions. With that context set, I can see a few ways of answering the question, depending on which conditions are allowed.
One way of interpreting the question is "if the box contains photons, and I vibrate one of the walls, can I observe a consequence of the vibration at the other wall". The answer here is "yes". If you add or remove momentum to photons incoming at a wall by moving the wall, these photons will exert a corresponding reduced or increased pressure on the opposite wall, which could be measured. However, it is the same photons, under low energy conditions in vacuum, that propagate the information. This is in opposition to what we usually mean by "sound", in which, for example, the molecules that vibrate in your ears for you to hear are not the molecules that have been vibrated by a speaker. The momentum of the initially vibrated molecules is transmitted from molecule to molecule until they reach your ear.
A second way of interpreting the question is "if the box contains photons, and I vibrate one of the walls, will I influence photons other than the ones I directly touched with the wall". At low energies, photon/photon interactions in vacuum are practically non-existent. You would need very energetic photons to create any significant interactions. This has been mentioned in the other answers.
A variant of the second way of thinking would be to allow matter in the box. Indeed, one way to increase photon-photon interactions is to add some specific non-linear materials, which couple photons to each other. This is the basis for the field of non-linear optics. In that case, under specific conditions, you could transmit information through photons inside the material that would propagate by photon/photon interactions, but the photons interact with the help of the atoms in the material. A very technical paper that describes these phenomena in your context can be found at https://arxiv.org/pdf/1205.6500.pdf.