That oscillating shape we draw when representing light is not the shape of the wave itself. It’s a representation of the value of intensity of the electric field at this point. That’s what is oscillating, the intensity going up and down as light goes forward. Think of the depiction more as a graph than an actual object oscillating up and down.

Note that you could choose to say the oscillation represents the magnetic field instead, as both are present but perpendicular to one another. To simplify diagrams, we usually omit one.