No, it would not.

One starting point (in fact a postulate) of special relativity is that the speed of light in vacuum is the same for every observer, regardless of the motion of them or the source. Sound does not fulfil this requirement for a number of reasons:

  • Sound waves need a medium in which to propagate. The speed is different depending on the medium, and there is no sound in space. Even with air it varies according to pressure and humidity, though it is roughly constant for many practical purposes (about 330 m/s).
  • The speed of sound is fixed relative to the medium. Therefore, the observed speed depends on the motion of the medium (e.g. wind).

The invariance of the speed of light relates to the lack of a required medium, and it is not unique to light. It applies to all forms of electric and magnetic fields (light being a special case), the strong nuclear force, and supposedly gravitation. However, it is the same c in every case; relativity would collapse if we found something with an invariant speed different from c. It would be interesting, though.

Considering Pender's writeup, I might add that evolution has produced vision several times independently throughout the history of Earth. It is expected to happen in a world dominated by light.

On a final note, you could imagine that the people discover or invent something that is faster than the speed of sound in air. For instance, the strings in m_turner's example may have a rather high speed of sound along them. It is twice the length times the sound frequency, from some very basic wave physics. I believe the people would know this, if they were ready to develop relativity. If you take something like the string in a properly tuned guitar, you get wave speeds in considerable excess of 330 m/s.