I think the particles that move up and down in a wave are the particles of a wave propagating through a medium, in which case the constituents of the medium make an up-down movement which is passed to the nearby constituents.
The waves that correspond to radiation are totally different in Nature because they move through the vacuum. They don't propagate through a medium (of course they can move through, for example, a medium like glass, but the e.m. wave is passed on through the vacuum). Then how can this up-down movement of radiation be passed through empty space if there is no medium to which the up-down movement can be passed to?
A long time ago it was thought that the aether was the medium for those waves, but now we look at it in an entirely different way.
The answer is that the e.m. wave is built out of a magnetic and electric part. A (periodically) changing electric field causes a changing magnetic field which in turn causes a changing electric field, in such a way the wave is propagating (this happens in a vacuum as well as in media). A wave built only out of a varying electric field does not exist.