The successive emergence of SARS-CoV-2 mutations has led to an unprecedented increase in COVID-19 incidence worldwide. Currently, vaccination is considered to be the best available solution to control the ongoing COVID-19 pandemic. However, public opposition to vaccination persists in many countries, which can lead to increased COVID-19 caseloads and hence greater opportunities for vaccine-evasive mutant strains to arise. To determine the extent that public opinion regarding vaccination can induce or hamper the emergence of new variants, we develop a model that couples a compartmental disease transmission framework featuring two strains of SARS-CoV-2 with game theoretical dynamics on whether or not to vaccinate. We combine semi-stochastic and deterministic simulations to explore the effect of mutation probability, perceived cost of receiving vaccines, and perceived risks of infection on the emergence and spread of mutant SARS-CoV-2 strains. We find that decreasing the perceived costs of being vaccinated and increasing the perceived risks of infection (that is, decreasing vaccine hesitation) will decrease the possibility of vaccine-resistant mutant strains becoming established by about fourfold for intermediate mutation rates. Conversely, we find increasing vaccine hesitation to cause both higher probability of mutant strains emerging and more wild-type cases after the mutant strain has appeared. We also find that once a new variant has emerged, perceived risk of being infected by the original variant plays a much larger role than perceptions of the new variant in determining future outbreak characteristics. Furthermore, we find that rapid vaccination under non-pharmaceutical interventions is a highly effective strategy for preventing new variant emergence, due to interaction effects between non-pharmaceutical interventions and public support for vaccination. Our findings indicate that policies that combine combating vaccine-related misinformation with non-pharmaceutical interventions (such as reducing social contact) will be the most effective for avoiding the establishment of harmful new variants.