How do electromagnetic waves propagate even if there is no medium?

They are electric and magnetic fields. A changing electric field creates a magnetic field and then a changing magnetic field creates an electric field. This process continues on in empty space. The fields propagate at the speed of light. 

Keep in mind that while we can describe electromagnetism with a wave equation, it is NOT like a sound wave, water wave, or other types of classical waves so it’s not surprising that it doesn’t need a medium like those waves.

Electromagnetic waves, like all elementary particles, have both wavelike and particle aspects. The particle associated with electromagnetism is the photon. In this case, it’s best to think of electromagnetic waves as traveling particles. Just like other particles, they need no medium.

Another way to look at it is based on special relativity. Light travels at a constant velocity regardless of the motion of an observer. If light had to travel through a medium (like the old concept of an aether) that would establish an absolute frame of reference in violation of relativity.

.An electromagnetic wave is an electric field and a magnetic field perpendicular to each other that propagates in directions perpendicular to to the 2 fields. No medium is required. 

No.