The last time I was on a bus travelling out east, there was a Mennonite man who was talking about vaccines with the bus driver. I was surprised to overhear that he was pro vaccine, and that there isn't anything in his belief system that mandates he be anti-vaccine.
So I don't know what drives the anti-vaxx view for Mennonites, but from what this man was saying it doesn't seem to be something that is inherent to being a Mennonite (like blood transfusions for JWs).
I live in a region with a lot of Amish and Mennonite groups. As I understand it, there's no central authority, but each community can make their own rules. Also, he may have been following his own instincts, independent of his sect.
I wonder if it's simply the fact that there really isn't anything driving them to get their kids vaccinated rather than a particular religious conviction. In Ontario, the old-order Mennonite and Amish groups have separate schooling for their kids and aren't integrated into the medical system here (not even being a part of our public health insurance system). Your family doctor and public health agency (through the schools) are the avenues the vast majority have to vaccination and so being apart from that, the old-order families would need to make a special effort to get vaccinated above and beyond what most people need to do.
So I don't know what drives the anti-vaxx view for Mennonites, but from what this man was saying it doesn't seem to be something that is inherent to being a Mennonite (like blood transfusions for JWs).