You'll never actually get studies "proving" this in a scientific sense, unfortunately, because there's really no way to design an ethical experiment. You can't force one group to take homeopathic treatments vs another group receiving medical care, just as you can't force one group to take vaccines while another doesn't.

That being said, there are longitudinal studies demonstrating the rates of autism amongst vaccinated individuals and I wouldn't be surprised if someone had done a similar study regarding medical outcomes for people who take homeopathic treatments. It's just that scientists will never call that "proof" because the bar is very high, and media will always conflate that with "the jury is still out who knows what's right" because they're profit hungry bastards.