The only credible research I've seen showed a possible connection to intestinal bacteria getting out of control after vaccination in some young children, with the advice to simply spread the initial two dozen or so recommended vaccinations over a slightly longer period of time (I think it was 48 months), with prioritization given to highly infectious and deadly diseases (e.g. meningitis).
There are also a bunch of "non-medical" ingredients in some manufacturer's vaccines that are prescription drugs (statins, etc.) used to "bootstrap" the vaccines but that are not approved for use in children otherwise. Some localities do enforce requirements on such ingredients in children's vaccines while others don't... I'm not aware of any specific research into the side-effects of such ingredients, but (as an example) our son's pediatrician avoided given vaccines containing unapproved ingredients to children out of simple caution.
Ultimately I think the biggest problem is that both "sides" are demonizing the other, with "pro vaccine" people calling anyone who has questions or fears about vaccines an idiot, dangerous, etc. and governments providing a liability shield to vaccine manufacturers and forcing parents to give their children more and more vaccines on an accelerated schedule, often with little or no notice. As an example, we were told our son's vaccinations were up to date at the beginning of the last school year only to be told 8 months later he needed another vaccination or he would not be allowed to continue going to school. Getting a notice from the government saying "do this or else" is hardly a way to build a trusting relationship. And experts not talking openly and freely with those that have concerns forces those with concerns to talk with the "alternate experts" that are willing to fill their heads with their agenda.