Can I apply Fourier Transform to a non-time-indexed signal? Say I have a signal that is not x-indexed. That is, the x-axis of the signal is the distance traversed by car and the y-axis is the heading direction of the car at the corresponding distance.

Can I apply the Fourier Transform to this signal?
If so, what is the physical meaning of this transformation? I believe that the horizontal axis is no longer frequency any more. What is it in this case?
Yes you can. The unit of the "frequency" axis after the transform will be $m^{-1}$, and is known as spatial frequency.

For example, if there is a strong peak in the Fourier transform at $3.10^{-4} m^{-1}$, it means that your original curve exhibits a strong pattern that repeats at a scale of every $3.3km$, and from that you could infer that maybe the signal was recorded from a vehicle doing laps at the Monaco Grand Prix. The harmonics of this spatial frequency would contain a "signature" of the shape of the circuit.

A practical application of this is handwriting recognition - looking at shapes in the Fourier domain yields representations invariant to scaling, rotations, or more robust to deformations than the original data.