This definitely cuts to the heart of scientific method, documentation of research, and brings up the issue of political and social attitudes toward technological and scientific "experts".
There is as yet no way to "prove a negative" - to absolutely prove without any question that the mercury containing additives in these vaccines never caused harmful effects. What the researchers report in the medical and science literature is often based on statistics of test and control groups. Thus, in the 285 hits, going back to 1998, in Science Citation Index for the search: Topic=(autis* and vaccine*), the titles very often say something to the effect that "evidence does not support" a link, or "no conclusive evidence of a link has been found", etc.
But short of investigating - on our own - every report, and combing it for inconsistent or careless documentation, false or careless reasoning, mis-statements of other people's claims or research, etc., there will be no way to know who is telling the truth.
The best we can reasonably do is examine samples of the scientific literature, follow the money trail to see who is funding the research, be on the lookout for inflammatory or other opinion based statements, and in general resolve not to just take someone's word - even if they are a Nobel prize winner.
The same would go for the people who accept a link - How do they know? How many people did they study out of how many total cases, and how did they select their sample(s)? Do they clearly state their credentials, or otherwise prove - maybe by brilliant and well-documented arguments - that they know what they are talking about? We librarians will always need to ask "who says so?", because we have too often had experts (including ourselves!) tell us stuff that turned out later not to be 100% accurate.
If a website doesn't give references or even say where its author is working, I tend to assume the information is suspect.