In this paper, we developed a SIRV epidemic model based on a vaccination game, incorporating vaccination dynamics and data memory effects using the Caputo fractional derivative. This approach effectively captured the nonlocal and power-law characteristics of influenza transmission. We confirmed the model's biological well-posedness, proved the uniqueness and existence of solutions, and analyzed stability. Furthermore, we established global Ulam-Hyers stability. The results showed that the epidemic incidence depended on the number of reproductions in the system. Through the Grünwald-Letnikov method, we developed the numerical simulations. We validated our theoretical findings and provided insights into the impact of vaccination on influenza progression. Our simulations revealed that strategic vaccination decisions were influenced by individual perceptions of the benefits and costs to achieving control of the influenza disease.