Ensuring the rational and orderly circulation of medical supplies during a public health emergency is crucial to quickly containing the further spread of the epidemic and restoring the order of rescue and treatment. However, due to the shortage of medical supplies, there are challenges to rationalizing the allocation of critical medical supplies among multiple parties with conflicting interests. In this paper, a tripartite evolutionary game model is constructed to study the allocation of medical supplies in the rescue environment of public health emergencies under conditions of incomplete information. The game’s players include Government-owned Nonprofit Organizations (GNPOs), hospitals, and the government. By analyzing the equilibrium of the tripartite evolutionary game, this paper makes an in-depth study on the optimal allocation strategy of medical supplies. The findings indicate that: (1) the hospital should reasonably increase its willingness to accept the allocation plan of medical supplies, which can help medical supplies allocate more scientifically. (2) The government should design a reasonable reward and punishment mechanism to ensure the rational and orderly circulation of medical supplies, which can reduce the interference of GNPOs and hospitals in the allocation process of medical supplies. (3) Higher authorities should strengthen the supervision of the government and the accountability for loose supervision. The findings of this research can guide the government in promoting better circulation of medical supplies during public health emergencies by formulating more reasonable allocation schemes of emergency medical supplies, as well as incentives and penalties. At the same time, for GNPOs with limited emergency medical supplies, the equal allocation of emergency supplies is not the optimal solution to improve the efficiency of emergency relief, and it is simpler to achieve the goal of maximizing social benefits by allocating limited emergency resources to the demand points that match the degree of urgency. For example, in Corona Virus Disease 2019, emergency medical supplies should be prioritized for allocation to government-designated fever hospitals that are have a greater need for medical supplies and greater treatment capacity.