Edison effect in vacuum
In Edison effect in vacuum, is the current moving in the form of
  actual electrons or magnetic fields?


The current that Edison was observing was due to an electron current with electrons moving through a vacuum. The effect in vacuum tubes is from one side of the tube being heated to thermally excite electrons off of a metal plate, where they can transfer charge to another metal plate. So to answer your question the charge was due to actual electrons.

This is different than the fields Gauss was noticing, because he was experimenting with capacitors, the displacement current is due to electric and magnetic currents changing in the capacitor.