Hi OP
I got pulled into a lot of antivax stuff with my first DC. It really alarmed/worried me because before having children, I'd never heard anything bad about vaccines ever, I had only heard of them as this wondrous lifesaving thing. So being suddenly told that they could harm my baby was very worrying and it seemed that the more I "looked into it" the more scary stuff I found. (and of course feeling a bit anxious about my child having to have injections which would hurt - it's totally normal and natural to feel worried about this by the way).
I will cut a long story short and say that after some initial alarm/anxiety and worry I started to notice indiscrepancies and odd things about what people were saying in terms of the dangers of vaccination.
One thing which made me stop and question them was when someone encouraged me to print out a letter for the nurse/GP to sign inviting them to take "full responsibility" for any side effects of the vaccine. I realised immediately that nobody would sign such a document and nor would I expect them to. That's when I realised that the antivaxxers were protraying this idea of a totally 100% risk free vaccine - which does not exist. Everything has a risk. Feeding your baby food has a risk of an allergic reaction. Having your child undergo surgery has a risk. Calpol has a risk. Teething gel has a risk. Avoiding vaccines carries a risk. In fact, vaccines are some of the healthcare with the highest hurdles to jump to prove that they have a low incidence of side effects, specifically because they are given to healthy people. When you have surgery then it's usually to correct some problem that is already happening so we accept some side effects and risk as a trade off for that. When we take medicine like antibiotics again this is to correct something so we accept some side effects as a trade off. Vaccinations are given to healthy people and the amount of side effect people will tolerate is extremely low. And contrary to what the antivaxxers say, the medical profession are fully aware of and talk about the potential side effects, they will even advise if there is anything you need to watch out for - but the serious side effects, if they happen, typically happen immediately which is why you wait a short time in the GP surgery after a vaccination (and you can ask to wait longer if you are feeling anxious). The antivax crowd made it sound like "big pharma" are forever pushing more and more vaccines and want to give them unnecessarily - but if that was true, why are the covid vaccines now being phased out? Why have they phased out TB, and smallpox? They are also planning to take Polio off the schedule (except that it has currently unfortunately been halted in the progress towards eradicating the disease). "They" don't want to give unnecessary vaccinations - only the ones for which there is a proven benefit where the benefits outweigh the risks. And believe me, the balance required for this is very high.
Another thing which made me stop and question is the idea that vaccines contain "dangerous" ingredients, like heavy metals. But I also knew that for example the advice during pregnancy is to eat no more than 4 cans of tuna per week because of the mercury. So if it is safe to eat up to 4 cans, then there is a safe level of mercury to consume and an unsafe level. I realised that vaccines must contain absolutely miniscule amounts, and I'm willing to make a trade off for tuna just because I like the taste of tuna. But a vaccine actually protects my child from a dangerous disease - the trade off is extremely clear. Why not, if concerned about heavy metals, look at prioritising reducing other, less useful sources in order to free up some capacity to cope with the very miniscule amounts in this very useful medical intervention.
The antivax crowd are quite insistent that vaccines "don't work" and it's all just a con or a money making scheme by big pharma. There were all kinds of graphs thrown around which seem to show that drops in various diseases were caused by something else and not the vaccine - but there is an extremely clear and very very common example that you can see - chicken pox! In the UK, not a standard vaccine given, and it goes around every single school and nursery several times a year and almost every child gets it at some point, to the point it's unusual if you haven't had it and people even want their child to yet it young to "get it over with". In other developed countries, children get it generally with the MMR and nobody gets chicken pox. Honestly. Nobody. I live in Germany now, I have been here for ten years and I have come across I think one case of chicken pox.
Lastly they spread a lot of info about how the diseases that vaccines protect against are not that harmful - unfortunately this is a lie. While it's true that if your child catches measles, the chances are that on balance they will likely be fine, the problem with measles is not so much the risk to your child personally, the problem is that measles is extremely contagious, so when it is allowed to run freely through populations, vast vast numbers of people catch it and that means that a great number of them, even though a small proportion (this can be the same thing - think about 1% of 3,000,000 people - that's 30,000 people) will suffer severe, permanent side effects or even die. The burden being greater on the portions of the population who are more vulnerable - the sick, the elderly, the very young, those with poor housing conditions, those with predisposed genetic tendencies etc.
Unfortunately there are some families who are living in limbo with rare, undiagnosable conditions (generally genetic or autoimmune disorders) that science can't provide a nice, easy answer to. Antivaxxers have a simple and convincing argument - which is exactly what should alert you to the fact that it's not the right one, as well as all the many, many, studies and huge amount of evidence showing that autism, for example, is not caused by vaccines (and some other things). This is the worst part IMO. Selling answers to distressed people is immoral and cruel. Using them as a vehicle to spread distrust is despicable.
At the time I decided to vaccinate my eldest (several months behind schedule) I was still not convinced I was doing the right thing, but I felt that the risk was probably about the same from vaccinating as it was if I just let him go about in the world considering that he would probably not encounter the dieases anyway. The reason that I made the decision to vaccinate was actually because I thought, on balance, if the two options were probably the same then it made sense to go the way most people go because that way doctors would be more familiar with the risks we were taking, whereas if he ever had got a vaccine-preventable disease, doctors today would be unfamiliar with those and that might pose a problem in itself. I was very anxious when the vaccines were being given and for a few months afterwards in case any side effects showed up (they didn't).
Fifteen years later I think I was wrong even then and I was taking a much, much, much smaller risk than I thought at the time in vaccinating him. I have vaccinated my younger children with no worries. The more I learn about where antivax rumours come from, the more I understand why they are so alarming and why they are also not to be trusted. I would recommend speaking to your doctor or nurse about your specific worries if you can, as they can often explain why the common antivax myths don't make sense to worry about.
There are quite a few people speaking out who have previously been hoodwinked by the antivax community now - I will send some links as I think it's interesting to hear what people say who used to be in this sphere and why they came out of it.