I can conceive of a particle existing in empty spacetime, but not a wave. A wave appears to me at least, to insist upon a medium for its very definition.
I understand that the 19C physicists postulated the lumineferous aether for this very reason, but abandoned it in favour of the electromagnetic field.
In the textbooks I've looked at, its always said that the physicists eventually understood that a wave does not necessarily require a medium. But it seems to me tht they abandoned the idea of a medium mechanically conceived (I assume because of the high prestige of newtonian mechanics, its status as a defining, fundamental paradigm which privileges our naive & intuitive understanding of mechanical phenonema) for a medium conceived in a a more general manner.
Afterall, this field permeates spacetime, and its state evolves through time. This is how I would naively conceive of a medium in a general sense.
I should add this is maybe more of a history of physics question, and I've added a tag as such to stop people getting upset.
Answer
What is colloquially called ''empty space'' isn't really empty - it is filled by the electromagnetic field and the gravitational field; it is called empty only because it doesn't contain (nonzero) matter fields.
The electromagnetic is the medium that carries electromagnetic waves, such as the air density field (colloquially just called ''air'') carries sound waves and the water density field (colloquially just called ''water'') carries water waves.
Indeed, electromagnetic waves are nothing else than propagating high-frequency oscillations in the electric fields, in precisely the same way as sound waves are propagating ohigh-frequency scillations in the pressure field of air (or any other mechanical medium), and water waves are propagating low frequency oscillations in the mass density field of water.
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