Electromagnetic (or light) radiation consists of changing electrical fields which generate a magnetic field which in turn generates and electric field. This process is very fast. In fact, electromagnetic waves propagate as fast as it is possible for anything to move.
Sound waves consist of waves in a physical medium. Big, fat, slow (from the point of view of light anyway) molecules have to move through space, bump into other molecules, and transmit the wave that way. It's a totally different process than EM waves. It's also much slower.
The reason that doppler effects are more pronounced with sound, is that the magnitude of the doppler shift depends on the velocity of the observer or source as a percentage of the speed of the wave. A vehicle moving at 60 miles per hour is going at about 10% of the speed of sound, but only 1/100,000 of 1% of the speed of light. The doppler effect for sound emitted from the vehicle will therefor be far larger than the doppler effect for light emitted by it.
