The COVID-19 pandemic has caused a worldwide crisis due to its rapid and extensive global spread. As the coronavirus is an RNA virus with a single strand, there are no specific medications, such as antibiotics, that can be used to directly fight against it. Therefore, vaccination and nonpharmaceutical treatments are the sole remaining strategies that have a significant economic effect in battling a pandemic. Currently, effectively managing this disease is a significant global concern. The mathematical model can assist in determining the optimal techniques for illness control and assessing their impact on disease dynamics. Inspired by the aforementioned considerations, we have examined a new mathematical framework in this study to analyze the transmission of the COVID-19 epidemic in India, taking into account public intervention and vaccination. The model comprises eight compartments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, isolated, hospitalized, and recovered. These compartments are represented by a system of eight interconnected nonlinear ordinary differential equations. An extensive analysis of the model was presented, which included the derivation of the disease-free equilibrium point, the calculation of the basic reproduction number using the next generation matrix, and the stability analysis of the disease-free equilibrium point. We examine the model by employing the Caputo–Fabrizio (CF) fractional differential operator. The propagation of COVID-19 was solved by solving the system equations using a numerical technique based on the Lagrange polynomial. Numerical simulations illustrating different fractional orders demonstrate the appropriateness of fractional derivative operators.