The COVID-19 pandemic has affected people's lives worldwide. Among various strategies being applied to addressing such a global crisis, public vaccination has been arguably the most appropriate approach to control a pandemic. However, vaccine supply chain and management have become a new challenge for governments. In this study, a solution for the vaccine supply chain is presented to address the hurdles in the public vaccination program according to the concerns of the government and the organizations involved. For this purpose, a robust bi-level optimization model is proposed. At the upper level, the risk of mortality due to the untimely supply of the vaccine and the risk of inequality in the distribution of the vaccine is considered. All costs related to the vaccine supply chain are considered at the lower level, including the vaccine supply, allocation of candidate centers for vaccine injection, cost of maintenance and injection, transportation cost, and penalty cost due to the vaccine shortage. In addition, the uncertainty of demand for vaccines is considered with multiple scenarios of different demand levels. Numerical experiments are conducted based on the vaccine supply chain in Kermanshah, Iran, and the results show that the proposed model significantly reduces the risk of mortality and inequality in the distribution of vaccines as well as the total cost, which leads to managerial insights for better coordination of the vaccination network during the COVID-19 pandemic.