The outbreak of mutant strains and vaccination behaviors have been the focus of recent epidemiological research, but most existing epidemic models failed to simultaneously capture viral mutation and consider the complexity and behavioral dynamics of vaccination. To address this gap, we develop an extended SIRS model that distinguishes infections with the original strain and a mutant strain, and explicitly introduces a vaccinated compartment state. At the behavioral level, we employ evolutionary game theory to model individual vaccination decisions, where strategies are determined by both neighbors' choices and the current epidemiological situation. This process corresponds to the time-varying vaccination rate of susceptible individuals transitioning to vaccinated individuals at the epidemic spreading level. We then couple the epidemic and vaccination behavioral processes through the microscopic Markov chain approach (MMCA) and finally investigate the evolutionary dynamics via numerical simulations. The results show that our framework can effectively mitigate outbreaks across different disease scenarios. Sensitivity analysis further reveals that vaccination uptake is most strongly influenced by vaccine cost, efficacy, and perceived risk of side effects. Overall, this behavior-aware modeling framework captures the co-evolution of viral mutation and vaccination behavior, providing quantitative and theoretical support for designing effective public health vaccination policies.