Humanitarian organizations commonly establish facilities to store emergency supplies in advance, ensuring a prompt and adequate response to non-climate and sudden-onset natural disasters. However, existing research on facility location and inventory pre-positioning does not adequately examine multiple sources of post-disaster emergency supplies, which are critical for an efficient response. Additionally, disaster managers encounter various uncertain parameters when making decisions, such as disaster characteristics, demand, transportation, and supply. To address these issues, we develop a novel two-stage distributionally robust optimization model to minimize the total expected costs in the worst case. The facility location and quantity of pre-positioned emergency supplies are decided in the first stage before the occurrence of disasters. In the second stage, emergency supplies that are pre-positioned in advance or purchased from spot markets, as well as in-kind donations, are distributed to affected areas. Two tailor-made column-and-constraint generation (C&CG) algorithms, consisting of an exact and an approximation scheme, are designed to solve the proposed model. A case study based on the data collected from the earthquakes in Pu'er City, Yunnan Province, China, demonstrates the superiority of the proposed model, and provides managerial insights for decision-makers. Furthermore, the effectiveness and efficiency of the proposed algorithms are demonstrated by the results of experiments conducted on randomly generated instances.