To be fair, it’s perfectly reasonable to be skeptical. Major corporations, research labs, and the medical community as a whole are not infallible in their work. It is a novel virus, and there are no known vaccines for any coronavirus. But the doubt, fear, and need for persuasion have no real connection to Donald Trump.
That Trump would prefer to have a vaccine approved before the election is obvious. But wanting approval fast does not distinguish him from the leaders of any other nation, all of whom have an urgent interest in seeing that their own institutions and manufacturers come up with the best COVID vaccine, not only as a matter of ending the crisis, but as a matter of realpolitik. (Nor does it make his incentives all that different from the leaders of certain blue jurisdictions, whose prolonged lockdown restrictions are in a sense premised on the imminent availability of a vaccine.)
The vaccine is not exactly “rushed” in any case. While approvals have been given faster than normal for new research, the fact is that the pandemic itself accelerates the research process. The number of research subjects is large enough to conduct what are normally years-long trials in only a few months.