Consider for a moment that the "antivaxxers" are correct, that there is enormous bias in our medical, pharmaceutical, and media establishments in favor of vaccine safety. Could you really expect a reliable source on the subject if this were the case? What proportion of people are willing to speak out and risk career and social suicide for being labeled an "antivaxxer"? It is a self reinforcing chilling effect - few people speak out, data does not surface, fewer people look for evidence of adverse events, and over time it becomes riskier and more difficult to convince people that something is amiss.
Meanwhile the internet is awash with anecdote of severe symptoms following vaccination, and particularly concerning are frequent reports of doctors who are dismissive and/or refuse to submit reports on adverse effects. And then there are occasional claims of specialists in various neurological or heart related disciplines who claim to be seeing an unprecedented spike in certain normally rare symptoms among vaccinated patients.
Yeah, they could all be lying or this could be the nocebo effect or it's an army of bots spreading antivaxx propaganda...or maybe you can't magically accelerate a 5+ year testing and evaluation schedule for a novel technology merely because the president says he wants a vaccine yesterday? Note that the clinical data from pfizer and moderna is not available to the public - we are essentially relying on the manufacturers good faith with respect to the safety data that lead to approval by the FDA (moderna still isn't approved, by the way).
https://www.bmj.com/content/375/bmj.n2635
https://www.bmj.com/content/376/bmj.o102