I really think that this is a NO-BRAINER! I really like the phrase, "There is no evidence to suggest/prove....". Of course this doesn't mean that there IS NO evidence that this is true! "The absence of evidence is not the proof of absence.".
that there is no evidence to suggest means that there is just that 'no evidence' . If the idiot you are quoting wants to argue that the vaccine is up to 50 times more likely to kill kids, it's up to him to provide the evidence and prove his case. Otherwise anyone can say anything and leave it up to the other person to prove them wrong .