In this paper, we analyze a vaccination control in a large population of agents for the Susceptible-Exposed-Infected-Recovered (SEIR) model. We model the problem of selfish vaccination as a mean field game in continuous time with a finite number of states. Our findings indicate that this game yields a unique and pure equilibrium characterized by a bang-bang strategy. Furthermore, we evaluate the overall cost of the mean field equilibrium and contrast it with the cost of the socially optimal vaccination strategy. The results of the study show that when individuals are left to decide whether to vaccinate or not, the mean field equilibrium stops vaccinating at an earlier time than the socially optimal strategy. This implies that promoting vaccination may be necessary to incentivize individuals to adopt the optimal behavior and reduce the overall cost of the mean field equilibrium.