Two years after COVID-19 vaccine rollouts began, COVID-19 vaccination rates in low- and middle-income countries (LMICs) continue to lag. Tracing individual country experiences with vaccine procurement in the early stages of the pandemic suggests that international supply chain failures caused initial delays. High vaccine hesitancy in the population and last-mile delivery challenges within LMICs were other possible limiting factors. This paper summarizes descriptive and experimental research on vaccine demand and supply in LMICs to evaluate these competing claims. The weight of the evidence suggests that external supply restrictions and internal distribution challenges (rather than vaccine hesitancy) appear to be paramount.