I understand that EM waves are oscillating electric and magnetic fields. But doesn't this mean that the wave itself contains charged particles that generate the fields?
No. An electromagnetic wave is a disturbance in the electromagnetic field which contains an electric field that varies in time accompanied by a magnetic field that varies in time, and the disturbance propagates. When the disturbance passes by, nothing is left behind, and within the field disturbance (what we call a wave) there are no charged particles or little magnets present at any time.
It depends. Electromagnetic waves are possible in vacuum and quite abundant. These do not involve any charge or current distribution. In a transparent medium, such as glass, em waves do involve charges which while moving in synchronisation contribute to the character of the waves. Plasma waves and surface plasma polaritons also involve charges.
No. Electromagnetic Waves do not actually 'contain' anything. Classically, you can describe them as perturbances of the Electric and Magnetic Fields in space that are in a certain relation ( such as: they satisfy Maxwell's equations and D'Alembert equation, aka the wave equation). An electromagnetic wave can be described as one ( or more ) photons in a quantum approach. Either way, they do not contain electrons. They, however, interact with electrons: the electric field of the wave causes a force on electrons , and if they are moving they will experience a Lorentz force because of the wave's magnetic field.
The canonical answer is no: the electromagnetic field is itself a kind of matter, distinct from what we usually percieve as matter in terms of particles, such as electrons, protons, etc.
The modern view mentioned above was not always generally accepted: there have been many attempts to model electromagnetic waves as movements of a special substance, called Luminiferous aether or simply ether, which would make these waves similar to waves in water. The popular discussion can be found, e.g., in Jules Vernes' From the Earth to the Moon (or Around the Moon - I don't remember which exactly, just that I was stunned to read it as a serious discussion).
@akhmeteli in the comments has pointed out that in QFT one may harder time differentiating electromagnetic field from electrons:
Do electromagnetic waves contain electrons? Well, this may be a stretch, but one can probably give an affirmative answer citing vacuum polarization.
Polarization of a medium
Finally, if an electromagnetic wave propagates in a medium, it causes polarization (in simplest case represented by the dielectric constant $\epsilon$), which is due to the additional field created by the electrons being displaced by the field. Again, strictly speaking, the field does not contain the electrons, but they are often inseparable.
Weirdly, they don't contain electrons, but are actually made up of photons - even those waves that are not in a frequency we know as visible light.
Here's a source from Nasa.gov -
Every body above zero Kelvin emits photons. Electromagnetic radiation is the sum of all photons emitted by excited subatomic particles. It is basically thermal radiation.
Electromagnetic waves are a special form of EM radiation. The synchronous and periodic acceleration of electrons on the surface of a conductor results in radiation of polarized photons. This radiation has a periodic maximum of photons - it is a propagating wave.
The point is that each photon is an indivisible particle, moves in vacuum with the same velocity c, and has an oscillating electric field component and an oscillating magnetic field component. Photons support themselves with their oscillating fields. They don't need any environment to exist. On the contrary, every matter on the way of the photon leads to the absorption of a photon by subatomic particles.