Nope... To the stationary observer, the car, travelling at the speed of light, would emit a photon of light, also travelling at the speed of light. The two would appear to travel together.

Suppose the car turned on its turn-signals: i.e., emitting light at 90 degrees to its path of travel: the light would travel at c, not at c times the square root of two, as one might expect from vector addition.

Suppose the car flashes its brake-lights: i.e., emitting light backwards. The light would travel at c, not at a speed of zero, as one might expect from vector addition.

Given these ideas, the Lorentz-Fitzgerald equations fall out.

Silas