When you regard electromagnetic radiation as being composed of particles instead of waves, there is no question. Indeed, the particle-wave duality is rather confusing, but the simple truth is that sometimes we must regard it is a particle, and other times a wave, depending on what we are doing.

Electromagnetic waves travel through a vacuum and any other physical medium slows or stops their progression. They do not use existing magnetic fields as a medium, though they do interact with them.

An electromagnetic wave is nothing more than an a propagation of a changing electric and magnetic field. these fields are always perpendicular to each other and travel in the direction of E x B, where E and B are vectors indicating the direction of these fields.

Its because these are electric and magnetic fields. These fields are just regions where forces are exerted on charged particles, they are not made of particles themselves in the same way that sound waves are made from air molecules.