In case of supply chain disruption following severe disasters, many supply chains tend to collapse and take a long time to recover. Resilient supply chain network design (RSCND) is an important research problem in supply chain management, which means that the supply chain can maintain continuous supply and quickly restore the supply capability in part destruction. Based on the limited distribution information of uncertain demand, a two-stage distributionally robust optimization (DRO) model with ambiguous chance constraint (ACC) is proposed to solve the RSCND problem under demand uncertainty and disruption scenario to provide decision support for planning the supply chain network. Finally, to verify the effectiveness and practicability of the proposed DRO model, we apply the method to a real case study in Wuhan, China, about designing a resilient RSC network to withstand disruption. By comparison and sensitivity analysis in numerical experiments, some management insights of industry decision-makers are obtained.