Childhood vaccines play a vital role in social welfare, but in hard-to-reach regions, poor transportation, and a weak cold chain limit vaccine availability. This opens the door for the use of vaccine delivery by drones (uncrewed aerial vehicles, or UAVs) with their fast transportation and reliance on little or no infrastructure. In this paper, we study the problem of strategic multimodal vaccine distribution, which simultaneously determines the locations of local distribution centers, drone bases, and drone relay stations, while obeying the cold chain time limit and drone range. Two mathematical optimization models with complementary strengths are developed. The first model considers the vaccine travel time at the aggregate level with a compact formulation, but it can be too conservative in meeting the cold chain time limit. The second model is based on the layered network framework to track the vaccine flow and travel time associated with each origin destination (OD) pair. It allows the number of transshipments and the number of drone stops in a vaccine flow path to be limited, which reflects practical operations and can be computationally advantageous. Both models are applied for vaccine distribution network design with two types of drones in Vanuatu as a case study. Solutions with drones using our parameter settings are shown to generate large savings, with differentiated roles for large and small drones. To generalize the empirical findings and examine the performance of our models, we conduct comprehensive computational experiments to assess the sensitivity of optimal solutions and performance metrics to key problem parameters.