This is a variant of the Prisoner's dilemma, where if everyone does what's in their immediate best interest then everyone suffers needlessly.
It would only be the Prisoner's dilemma, if it was always better not to get vaccinated, regardless of what others chose. But in the "vaccination game" you want to get vaccinated if nobody else is and you don't need to if everybody else is. So you want to do the opposite of the crowd. That makes it a Chicken game.