Does conduction require a medium

Yes. Conduction involves the transfer of thermal energy without any movement of the material medium. For example, when one end of a metal rod is heated, the atoms at that end of the red vibrate more vigorously (gain kinetic energy), and they will collide with the neighbouring less energetic atoms and transfer its kinetic energy them and so on. Without these particles, thermal energy cannot be tranferred through conduction. The only form of thermal energy transfer which does not require a material medium (i.e., the atoms and molecules) is infrared radiation.