Oh FFS, it is **** like this that makes me question anyone up on the hill these days.
Yes, Vaccines can have a small chance of minor side effects. No, there is little empirical data to suggest that vaccines have some disposition to causing serious side effects or "profound mental disorders." Yes, most infections are more likely than vaccines to ensure lifelong immunity. No, you are not better without vaccines anyway, as in some cases infections lead to secondary complications... some of them are severe (as in more severe than the minuscule chance a vaccine ended up causing a serious side effect.)
Rand Paul is probably speaking more from a mindset of freedoms to make the choice, but the quote makes it seems like he is talking about some large percentage chance of vaccines leading to "profound mental disorders." As of yet, there is little evidence to back up that case.