Is it possible to get the 56e to turn the computer on with a button press?

Not yet, but it can be implemented in PoKeys57E.

Does this mean that someday it may be possible?
Is it possible to manually code this in the 56?

Since the PoKeys56E devices have a limited amount of memory and we almost ran out of it, the firmware development has now been focused to PoKeys57 series. The PoKeys56 series firmware will receive only fixes or smalller feature updates.
Therefore, the wake on LAN is not in plan for PoKeys56E device, but we will take this into account while doing some network feature updates for PoKeys57 series.