Answer
There is a Doppler shift for light as well as for sound.
Work Step by Step
As the star rotates, assuming its axis doesn’t point straight at the Earth, one side moves away from us and the other side moves toward us. This shifts the received light toward longer and shorter wavelengths, respectively (so-called redshifts and blueshifts).
From the size of these shifts, astronomers infer the tangential speed of the rotating star, and from that, deduce the rotational speed.
