Light waves are composed of electrical and magnetic fields. Such fields can exist in a vacuum. The definition of a vacuum is simply a region which does not contain matter, but there can still be electrical and magnetic fields.
Confusion often arises because people think that waves need something to push against. They don't. A similar misconception occurs with the notion of rocket power. People often ask how a rocket can work in space because it's not "pushing" against anything in a vacuum. But the fact is a rocket is the only kind of engine that will work in space because it doesn't need anything to push against. The force of the engine sends the rocket in an equal and opposite direction just because it is there.
That's it. Same with electromagnetic fields and waves. Think of electromagnetic radiation as a wave/particle duality. If you need a way to conceive of how it can travel through a vacuum, think of it as a particle while it's in the vacuum.