You are asking how there can be fields in vacuum, even though as you say there are no electrons present to create the EM field itself.
You are confused because of the definition of vacuum. Even though vacuum in some context might mean sometimes the same as empty space, the fields still fill the fabric of spacetime throughout, that is, for example the EM field exist in all of spacetime everywhere.
Quantum field theory implies that the vacuum is full of virtual particles that emerge and quickly disappear. Those virtual particles make their impact on other objects - for example, they make the electromagnetism a little bit weaker at long distances (and stronger at very short distances) than what one expects from the classical Coulomb's law etc.
However, quantum mechanics implies that the vacuum corresponds to a very particular "state" - a vector on the Hilbert space - called |0⟩. It is completely unique and as empty as you can get. In particular, it is the eigenstate of the energy operator with the minimum allowed energy - essentially zero. (More precisely, the vacuum energy density is nothing else than the magnitude of dark energy but this energy only becomes sizable for huge, cosmological volumes of space.)
The uncertainty principle of quantum mechanics implies that when one measures things such as the intensity of the electric field in the vacuum - i.e. when the physical system is found in the state |0⟩ - one may get many random values. It is not allowed for the electric and magnetic fields to be exactly zero, much like a particle can't have a well-defined position and velocity in the quantum mechanics of one particle.
So even though the vacuum has a well-defined (minimal) energy and it is as low as we can get, so the vacuum is as empty as we can get, and there are no particular "atoms" or other particles sitting in the vacuum, there's a lot of activity going on in the vacuum which can be seen by the fact that the measurements of various things, such as the density of energy at a given point, will lead to random results that are not strictly zero.
Does vacuum (empty space) exist?
Electrons are excitation of the electron field. When an electron/atom emits a photon (EM wave), that EM wave travels as you say throughout vacuum. The photons (that build up the EM wave) themselves are excitation of the EM field that fills spacetime throughout. These fields exist everywhere even when there is no EM wave traveling. When there is an EM wave traveling, this means that the excitation of the EM field (photons) is propagating throughout spacetime.
So the answer to your question is, that even though in vacuum there might be no electrons (excitation of the electron field) present, the fields themselves are present everywhere throughout space. The excitation of the EM field (photons) can and do propagate independently of the source (electron, the excitation of the electron field), and these fields exist everywhere even in vacuum throughout space.
