A minority of children do react very badly to vaccines - that's why the government has a Vaccine Damage Payments Unit that pays out compensation to a very small number of children and adults who have suffered extreme reactions that have been proven to be a result of the vaccine.
If you read anti-vaccine literature, they will cite plenty of cases of children who have suffered vaccine damage but were not believed by their GPs, who told them that it was just coincidence and the illness had nothing to do with the vaccine. Which may well be true, of course, but what it means - according to the anti-vax campaigners - is that the adverse effects don't get reported back via the yellow card scheme, and the extent of adverse effects is therefore not acknowledged.
The anti-vax campaigners will also say that if you look at the leaflet that comes with each vaccination, it gives a very long list of potential side effects.
I think there's a difference between people who had their children vaccinated and genuinely believe that their children suffered as a direct result of the vaccine, and those people who just think vaccines are bad because they're not natural, or full of chemicals, or whatever, because that is just muddied thinking.
My own view is that, although it's easy to see that vaccines have saved millions of lives, we shouldn't completely suspend our critical faculties. We know that almost every pharmaceutical medicine has side effects, so it would be odd if vaccines were completely safe. The problem is, though, that if people don't believe in the safety of vaccines, they will stop having their children vaccinated, which will have catastrophic consequences.