If light moves from left to right along the x-axis, then the electric field line vibrates along what?? :(

All you can say about it is that it then vibrates perpendicular to the x-axis. But that could be up and down parallel to the y-axis, in and out parallel to the z-axis, or some of it in every possible direction perpendicular to the x-axis. We "polarize" the light when we want to pick out only one perpendicular direction and stop all the others.