I think that there are few Christians who believe that no Christian who doesn't vaccinate will get Measles.

Many Chrisitan's do believe that there's evil in the world. They believe that for some complicated reason that they don't understand sometimes God will allow evil to exist. God is supposed to be an agent who's actions a mere human can't predict with certainity.

According to that frame, if someone get's Measles it's because God wanted that to happen. If on the other hand a child dies because of adverse reaction of a vaccine that the doctor gave the child then the parent shares responsibility for that harm because he allowed the vaccination.

I also don't now how the example of Japan is supposed to convince any Christian that his supposed belief in God preventing Measles of believing Christians is wrong.

While we are at the topic of the effect of beliefs, I don't think there good research about how beliefs that people hold effect whether they get illnesses. Part of the reason is that most doctors who do studies about the immune system don't think that beliefs are in their domain because they study the body and not the mind.