I think the info here is more than sketchy.
It’s false disinformation. Fear Porn against the Vaccine.
As you inferred, there is no way a vaccine will immediately kill a breast fed child.
Indeed. There is a strong anti-vax movement (motivated by the belief among its leaders that the way to fight overpopulation is for more people to die from infectious disease); it was inevitable that anti-vax activists would try to dissuade people from getting the vaccine.