What I heard RFK say is that once upon a time, we had a few vaccines that did a very good job of protecting against some very nasty diseases. He said, in general, he's not against vaccines as a whole.
What he objects to is all the vaccines that they are trying to force on everybody now, some with no good reason or weak reasons, that have not really been put to the test as rigorously as they should or could. He also goes to point out how autism has sky-rocketed, as has obesity, and other things that did not exist ~40-50 years ago. He points to all the money flowing from pharmaceutical companies to the FDA, the chairman of this or that being on the board of safety oversight panels and all the conflicts.
He makes some really good points, if you can stand to listen to his damaged voice. He has also, at times, said some sorta whacky stuff that gives me pause.