While using the polarimeter to determine optical rotation, why can't we use polychromatic light? What would change?
The reason is partly historical and partly scientific. Most of the polarimetric data exists with the sodium yellow emission at 589 nm because sodium lamps were among the most convenient ones and remember electricity is rather new. Early light sources were either flames or sunlight. It was easy to generate sodium emission by introducing salt in the flame. If you are familiar with Raman spectroscopy, it was discovered using sunlight (heliostats) not lasers in the 1920s-30s. Today nobody can think of using sunlight for spectroscopy.
At a more fundamental level, optical rotation is a function of refractive index and refractive index is a strongly dependent on wavelength. You have to see the optics of a polarimeter, the question of using polychromatic light does not arise. You will see a number of prisms are used. Impossible to use polychromatic light.