Yes as with any autoimmune disorder, the gut microbiome is super important. My child is also 3 and we are cutting gluten and dairy completely to help her heal, also working with a nutritional therapist who specialises in healing child autoimmunity.
I have already healed my own autoimmune diseases by this method so I know it works.

I am not a doctor but in my experience there is no connection between milk and Vitiligo.