A friend of mine recently went through proceedings for a parenting matter, and the dispute about immunisations came up. The mother didn't want them done, the father did, and the father's lawyer simply told him to go and get them done, whether the mother agreed or not. Without confirming anything, it's my understanding the court put close to no weight on it, and the father ended up with 60/40 regardless.
If the dispute is about time spent with and not about shared parental responsibility, then the court is unlikely to put a great deal of weight on the medical decisions made by the respective parties because those parties have already agreed about how those decisions are made, even if they don't always agree about them. The thing that does come up in proceedings is whether the parties have consulted with each other about these matters.
If you don't talk to the mother at all and just do whatever you please, then you might have issues, but if you've tried to consult with the mother about the immunisations and she has refused to engage the conversation, then your next step is to do what you believe is in the best interests of the children. In this case, I would say that immunising your kids can definitely be defended before the court as what's in their best interests, and to put another spin on it, doesn't it seem unruly to not vaccinate them simply because you're scared about how it impacts your case? Time with your kids won't count for much if they end up infected with measles, eh?
Will you cop some heat from the mother? Sure, but what's she going to do about it? Unvaccinate them? Tell the court that her decisions about the kids' health is more important than your decisions about the kids' health? Tell the court she knows better than an overwhelming plethora of scientific evidence supporting vaccinations for kids?
If it were me, I would get them vaccinated whether the other parent agreed or not. The existence of proceedings certainly shouldn't pose a risk to the kids' health, and the mother will look nuts making a song and dance in front of the court about how terrible a parent you are for getting the kids vaccinated.
But that's just my view.