What would happen if a charge could travel at $c$? We know that an electron can move at sublight speed, but can we figure out what would happen if a charge moved at $c$, surfing the wave of the electric field it has produced? There does not seem to be anything that prohibits charged particles from moving at light speed. (If it sounds better, consider a Weyl fermion)
It would be pushed forward, repelled by its own field, but there are 2 possible scenarios depending on whether the speed of propagation of the wave can be exceeded or not.
 A: All fundamental particles with electric charge have experimentally been shown to have mass (and no fundamental particles have been modelled as Weyl fermions) so the question is theoretical only.
In classical electrodynamics there is nothing to prevent a source electric charge moving at velocity c (and Maxwell's equations for example don't include the mass of the source).
If an electric charge is moving at a constant velocity with respect to an observer, the observer would imagine the electromagnetic field created by the charge moving with the charge. It would have both electric and magnetic components that don't change in strength in time (with respect to the electrical charge). The 'shape' of the electric field can be imagined by taking the shape of the electric field created by an electric charge at rest and then length contracting in the direction of motion as per special relativity. This been covered on another post: Electric field generated by a point charge moving at the speed of light
Electromagnetic waves are only produced by an accelerating electric charge. If the charge was massless, special relativity theory implies it must always travel at speed c so producing electromagnetic waves (as defined in classical electromagnetism) from the charge would be an issue. (Incidentally, if an electric charge is passing through a medium, it is possible for it to move faster than the electromagnetic wave it produces - which is maybe the type of idea in the original question - an example is Cherenkov radiation. The electromagnetic wave will still move away from the charge and is zero strength in the direction of motion of the charge so there is no opportunity for the charge to 'surf' the wave). 
You could also examine the idea of a massless electric charge and it's implications in a quantum field theory - some of this has already been covered in the 'related' posts above.
A: The electron has mass. Particles that have mass cannot move at velocity c. There are no charged massless particles, thus no charged particles can move at c.
