In this paper, we analyze a mean field game model of SIR dynamics (Susceptible, Infected, Recovered) where players can vaccinate. This game admits a unique mean-field equilibrium: The equilibrium strategy of each player is to vaccinate until the proportion of susceptible players drops below some threshold and stop vaccinating thereafter. We also show that the vaccination strategy minimizing the total cost for the population is a different threshold strategy. This implies that, to encourage optimal vaccination behaviors, vaccination should be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ($N$) to the mean field equilibrium. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone.