Has anyone had a child with chronic ear infections and tried a milk-free diet to prevent them?

Yes: I've known a few kids to respond to this. This is not a scientific answer i'm giving you, but i've seen several occasions where children appeared to get better after all dairy (not just milk) was removed from the diet.

No evidence of that: There is no scientific evidence of milk intake causing ear infections. This is a longstanding tale without any data. People do get better sometimes after stopping milk. They also get better when stopping tomatoes, watching the moon rise, and having the democrats win a presidential election. In other words, it is usually coincidence, not cause and effect.