What oscillates in electromagnetic waves?
In a medium the wave 'signal' is transmitted by particle
polarisation and atomic scattering, at a rate subject to Fresnel's
refraction coefficient 'n'.
Between those particles, and in a vacuum, nobody really knows.
It used to be the 'luminiferous ether', but it's light transmitting
properties were 'removed' as a 'stipulation' of the Special Theory
of Relativity, (STR) which said motion and speed was only relative
and could not be 'in relation to any 'background''.
This was done to explain the constancy of light speed (CSL) with
respect to moving receivers. Many have pointed out it's logical
inconsistency (giving rises to paradoxes), but no alternative
theory explaining CSL has yet replaced it. The only one available
so far is the discrete field model (DFM), suggesting light changes
speed locally to 'c' at each receiver (complying with 'n'). This
allows the 'quantum field' (ex 'ether') back, resolves the
paradoxes and explains the anomalies, but the STR is a 'ruling
paradigm', which are almost impossible to change with current
scientific thinking.