Since the outbreak of the COVID-19 pandemic, many healthcare facilities have suffered from shortages in medical resources, particularly in Personal Protective Equipment (PPE). In this paper, we propose a game-theoretic approach to schedule PPE orders among healthcare facilities. In this PPE game, each independent healthcare facility optimises its own storage utilisation in order to keep its PPE cost at a minimum. Such a model can reduce peak demand considerably when applied to a variable PPE consumption profile. Experiments conducted for NHS England regions using actual data confirm that the challenge of securing PPE supply during disasters such as COVID-19 can be eased if proper stock management procedures are adopted. These procedures can include early stockpiling, increasing storage capacities and implementing measures that can prolong the time period between successive infection waves, such as social distancing measures. Simulation results suggest that the provision of PPE dedicated storage space can be a viable solution to avoid straining PPE supply chains in case a second wave of COVID-19 infections occurs.