It's a fun day when two different Republican presidential candidates get themselves in trouble about science. We've all heard the stories, Rand. That's why there's an issue. The question is not whether there are stories, but what you would advise to parents.
So here's what I think.
1) As a gambling man, I notice that lots and lots of people get these vaccinations, and almost none of them are the source of scary stories. The diseases seem to have much worse outcomes on average. So, a smart gambler takes the vaccine.
2) As an anecdote, I myself have been vaccinated against just about everything, and I'm just fine. I've even had vaccines for anthrax, small pox, and third world diseases that won't come across your desk unless you travel widely.
3) Furthermore, all the medical professionals I know -- including my favorite cousin -- tell me that they are aware of no evidence that these things are dangerous, and strongly recommend administering vaccinations to your children.
4) Meanwhile, not only will you be protecting your child if the vaccination works successfully, you'll be doing a good deed for other parents of other children as well. These things work much better if we all do it.
Now the fun part.
5) As a philosopher, I can tell you that the strongest argument is the argument from gambling. There's a lot of empirical evidence about outcomes. You're placing a wager of a sort, with your child's life and health as the stakes. If you view this as a wager, it's pretty clear what the smart bet is.
All the other arguments are suspect. My anecdote is of no use to you, because anecdotes are not data and your child's body chemistry is not the same as mine. In fact, even if we get to data, you still get no promises. Cabbage is widely administered to the population. Almost no one has any problem with it. My wife happens to be allergic to it. Weird body chemistry things happen all the time.
The appeal to medical professionals and scientists is an appeal to authority, which is an informal fallacy. This is their area of expertise, which makes it less dangerous, but it's still no guarantee of truth. The fact is that the best they can tell you is that they have no evidence, yet, of any connection. That's an argument from ignorance, which is another informal fallacy.
The final argument is an appeal to ethics, but ethics doesn't have a lot of clear objective standards. The only place you find objective standards in ethics is virtue ethics. You can show that courage is objectively a virtue, because no matter what your goals are, being courageous will (always or for the most part, as Aristotle says) help you achieve them. Vaccination is a virtue on this account: always or for the most part, it will lead to the best outcomes for your child. Vaccination is the virtuous thing to do just because it passes the gambler's test.
The ethical argument that you should take the risk to help other peoples' children, however, is suspect. It's not clear that there's a virtue involved in risking your child to save other peoples' children. Any claim that there's any sort of duty to do it is not objective: now we've left virtue ethics for what is called "Deontology," and nobody really agrees about what roots duties. It is not clear to me how you would ground any duty that required a parent to risk their child's life or health for any reason.
So, what should you do? Vaccinate your children. It's virtuous, and it's the smart bet. Don't let anyone tell you that it's not a risk, though, or that you're stupid for worrying about it. There's still a lot we don't know.
UPDATE: Speaking of Republicans seeking the nomination, Dr. Carson is a pediatric surgeon by training and his opinion is to vaccinate.