I'm sure it's possible, but that means effectively that said person is now an undocumented trial participant. They have gotten injected with an unapproved medical product. That's a risk they've taken in order to get in early. The physician that administered it has also committed a crime that can cost them their license to practice.
There are easier ways for people in power to get vaccines. In China, the government has granted exceptional permissions to vaccinate a number of people with a vaccine that has not yet completed phase 3 trials. The list of people who qualify for that vaccine is said to be several million people long. It's much easier to get on said list in China than it is anywhere else. A number of other governments have also made exceptions for political leaders and such, treating them with non-approved medication or vaccines. For example, the sheikh of Dubai has been vaccinated with the Sinopharm vaccine before approval (in a very public display of faith in said vaccine, in order to promote vaccination to the population). Some countries are saying they won't bother waiting for phase 3 trials to end on some products and will make vaccines generally available based on current data. This is the case for Russia (with Sputnik-V) and Morocco (with the Sinopharm vaccine). I consider it highly likely that their heads of state are vaccinated under an exceptional authorization.