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Monkeypox (MPX) is a viral zoonotic disease that was endemic to Central and West Africa. However, during the first half of 2022, MPX spread to almost 60 countries all over the world. Smallpox vaccines are about 85% effective in preventing MPX infections. Our objective is to determine whether the vaccines should be mandated or whether voluntary use of the vaccine could be enough to stop the MPX outbreak. We incorporate a standard SVEIR compartmental model of MPX transmission into a game-theoretical framework. We study a vaccination game in which individuals decide whether or not to vaccinate by assessing their benefits and costs. We solve the game for Nash equilibria, i.e., the vaccination rates the individuals would likely adopt without any outside intervention. We show that, without vaccination, MPX can become endemic in previously non-endemic regions, including the United States. We also show that to “not vaccinate” is often an optimal solution from the individual’s perspective. Moreover, we demonstrate that, for some parameter values, there are multiple equilibria of the vaccination game, and they exhibit a backward bifurcation. Thus, without centrally mandated minimal vaccination rates, the population could easily revert to no vaccination scenario.