I have heard the theories that vaccines are one of a number of things contributing to higher rates of immune disorders, allergies, asthma and so on.
The thing is that in modern life you can avoid many of the things people worry about contributing to this kind of thing - food additives, plastics, sources of heavy metals, unnecessary skin products, etc - but realistically, you can't avoid everything. It's likely that in daily life you make decisions to expose your child to aspects of these things, pollution for example, perhaps you take public transport because this is greener than driving a car, but by standing at the bus stop DC are exposed to diesel and other traffic fumes.
Vaccines, IMO, are one of those trade offs you can and should make even while being concerned about the ingredients in them. If you look into it, the amounts of those ingredients are absolutely miniscule, less than you'd obtain by standing near a road, for example, and the benefits of vaccination (both to the individual and community) are huge, comparable with the impact of taking public transportation instead of driving a car.
I absolutely believe you that your common-or-garden "antivax" or "vaccine sceptic" parent is not doing anything except repeating what they believe to be true, not for personal gain, not for any reason except for concern for others to hear what they have heard. I get that. But. It is absolutely not true that nobody has anything to gain from spreading antivax rhetoric.
Firstly, not everybody in the alternative health industry is honest. There are people and companies in that field who really want a piece of the Big Pharma pie/profits, and they aren't above playing dirty to try and turn people towards their products. Think about it - they don't have the platform or the budget of the big pharma groups, so they play underhand. Ironically, it's not dissimilar to the tactics of infant formula manufacturers, a group many antivaxxers/natural health fans absolutely despise.
There have also been proven instances of influence groups using topics like vaccines/alternative health/clean eating in order to identify groups of people who are easy to convince of certain things, and then using that platform/group to push a particular political ideology in order to influence an election or other public event. Now that really does sounds crazy/conspiracist, yet it's happened.
Look up "David Avocado Wolfe" - just one example that springs to mind. I can't remember if he has done antivax stuff but it's the same kind of thing, and antivax gets used in the exact same way.
It's scary TBH.