The emergence of multiple strains of SARS-COV-2 has made it complicated to predict and control the COVID-19 pandemic. Although some vaccines have been effective in reducing the severity of the disease, these vaccines are designed for a speciffic strain of the virus and are usually less effective for other strains. In addition, the waning of vaccine-induced immunity, reinfection of recovered people, and incomplete vaccination are challenging to the vaccination program. In this study, we developed a detailed model to describe the multi-strain transmission dynamics of COVID-19 under vaccination. We implemented our model to examine the impact of inter-strain transmission competition under vaccination on the critical outbreak indicators: hospitalized cases, undiagnosed cases, basic reproduction numbers, and the overtake-time by a new strain to the existing strain. In particular, our results on the dependence of the overtake-time on vaccination rates, progression-to-infectious rate, and relative transmission rates provide helpful information for managing a pandemic with circulating two strains. Furthermore, our results suggest that a reduction in the relative transmission rates and a decrease in vaccination dropout rates or an increase in vaccination rates help keep the reproduction number of both strains below unity and keep the number of hospitalized cases and undiagnosed cases at their lowest levels. Moreover, our analysis shows that the second and booster-dose vaccinations are useful for further reducing the reproduction number.