Question

What will happen if we use white light instead of sodium light?

Answer 1

You can’t. The Michelson Morley experiment requires measuring the position of interference fringes. With white light they would all overlap and you wouldn’t be able to determine the location of individual fringes.

What you could do is repeat the experiment with lots of different frequencies of light. Maybe someone has done that, but it seems a pretty pointless experiment - the frequency that was chosen for the original MM experiment was arbitrary, and there is no scientific reason to expect the experiment to work differently with a different arbitrary frequency.