I absolutely will not pass judgement on any of your personal choices, but to discuss this properly I need to be frank - homeschooling, natural medicine and the anti-vax movement are controversial.
You are exercising your freedom to make your own choices, which in itself is very laudable. Your decisions differ from those of the majority, which also is no bad thing. I also believe that the desire to follow our peers can be a hindrance to our enjoyment of life as it teaches us to do what others expect us to do instead of what we actually would like to.
Still, you have to accept that going against the flow of the majority is difficult. It is comparable to swimming upstream, or walking alone through a crowd heading in the opposite direction.
The majority of people accept the weight of scientific evidence on conventional medicine. Many of that majority realise there are concerns such as industry interference, but still go along with whatever is currently recommended, including vaccinations. Likewise, a majority of people send their children to school with many accepting that it isn't a perfect system but still believing that any exposure to negative elements may in the long run be beneficial life experience for dealing with the negative aspects of adult life.
Some of the hostility and suspicion that you encounter along the way is because some people find it very hard to justify their own choices when they haven't thought them through properly. Hearing you talk about your decision and the thought you put into it challenges them because they haven't thought about it, and that makes them uncomfortable. Also, as I previously stated, there are others who have thought about these things but still just "go with the flow" despite minor doubts about the school or healthcare systems. Your choices also challenge them, perhaps invoking a degree of guilt in themselves for suppressing their own doubts.
You must know and accept this if you are going to follow any "alternative" life choices, because while these kind of choices are evidence that you are in control of your own family's life, you cannot control anybody else's life nor their feelings. So if others do take issue with them that is largely out of your control.
If you stopped reading at this point you could quite easily walk away with an inflated self-belief that you are right and everybody else is wrong. I don't feel that is the whole picture you should walk away with. You also need to think about how you present these choices to others - in fact, you should consider whether you should talk about them with others at all.
"[Our family] don't like talking about Catholicism, homeschooling or natural medicine or midwifery much either, so maybe we don't have a ton in common, but it seems like vaccinations are used as a scapegoat for their absence."
I would imagine that you have loads in common with them! First of all, you are family, and you have a ton of history together. Secondly, you have kids, and there are countless family activities that you could enjoy together. You could be making new memories with your extended family - things that you and your kids could be talking about for years! Instead it sounds like you want to talk to them about your decision to homeschool or your homeschooling experiences, your decisions on healthcare, and your faith - things that you know make you different from them! And then you declare that you have nothing in common.
Just as some people avoid talking about things that challenge their choices, others constantly talk about the decisions they have made because they have to continually convince themselves as much as others that they made the right decision. If you say that isn't the reason you talk about these things then fine - but not talking about them will be a demonstration of your confidence. File them alongside all the other personal, private choices that we make but don't need to discuss with everybody else.
In conclusion, my advice to move forward in your relationship with your extended family is to try and cut down or even eliminate talking about the things on which you don't agree and likely never will.
My advice to tackle your immediate question of "how to tell" if they are avoiding you over the vaccination issue, or anything else would be to contact them and invite them over. If they decline, then ask why.
If they don't directly give a reason but you still suspect it is the vaccination issue, say:
We can't help but feel you've been avoiding us recently, which makes us really sad. Our kids miss your kids! Please, if there is a reason why you might be avoiding contact we'd like to know so that we can try and fix things.
If they respond to this, or if they come right out and say from the start it is the vaccinations, say:
I won't go into the details behind our decision not to vaccinate our children, because I respect that you may have a different opinion. In short, we don't have confidence in vaccinations, but if you do have confidence in them then you must believe your vaccinated kids are safe, so there's really no reason why our kids shouldn't play together. Your kids can't catch anything from ours because you've immunised them, and our kids can't catch anything from yours for the same reason.
I think this is the best approach because (i) it does not compromise your own principles, (ii) it respects theirs, and (iii) to a degree it also tests both your beliefs.
You may be right in your beliefs, but nobody likes people who are insufferably smug. I'm not insulting you - I cannot believe that anyone who puts as much thought as you have into your children's welfare would not be a decent person. But perhaps your enthusiasm about these things has overcome your ability to see how others perceive that enthusiasm. If you want to make different life choices and still be liked and accepted by others, you need to keep a little back.