Vaccination plays an essential role in the fight against Coronavirus Disease 2019 (COVID-19). The currently insufficient vaccine production capacity makes it difficult to balance supply with demand, which has led to a contradiction between command demand and limited supply. According to analysis based on game theory, the attributes of COVID-19 vaccines vary with supply strategies formulated by vaccine-producing countries. This means that vaccine-receiving countries can only be motivated to prepare operable vaccine distribution plans through the supply of COVID-19 vaccines as global public goods. The rational distribution of global public goods must be guaranteed by a global supply institution system. To that end, Elinor Ostrom’s eight design principles provide a basis for designing such a global supply system. This paper proposes a nested institution solution for guaranteeing the global supply of COVID-19 vaccines based on the design principles, which include clearly defined boundaries, proportional equivalence between benefits and costs, collective-choice arrangements, monitoring, graduated sanctions, conflict-resolution mechanisms, minimal recognition of rights to organize, and nested enterprises. To win this global fight against COVID-19, COVID-19 vaccines must not only be treated as global public goods, but countries must also be urged to coordinate cooperation in global institutional design, thus ensuring that COVID-19 vaccines can truly benefit all mankind.