The one good thing that Trump did as president was to authorize by executive order the development of the vaccines at warp speed to begin saving lives. Today's vaccines are so far "experimental". Of course, he knew that at the get-go. Nevertheless, they have been deemed to be safe and they do work.
Trump, his family, Giuliani, Chris Christie, and their cronies were among the first to be vaccinated -- along with Tucker Carlson, his wife and kids. They may all be jerks, but they're not fools.
Generally, it can take months or even years for the FDA to approve a vaccine. However, when the country is hit with a plague and people are dying by the thousands, a safe "experimental" vaccine is still the best, really the only way to combat the virus. Every indication is that the FDA will be officially approving some, and eventually all, of the vaccines very, very soon.
However, it is critically important that everyone get vaccinated right away to stop the spread. Delaying a decision only results in more loss of life.