The COVID-19 pandemic broke out and the global logistics industry suffered severe losses; therefore, the Fuzzy FMEA-AHP (Fuzzy Failure Mode and Effects Analysis-Analytic Hierarchy Process) method is proposed to analyze the failure reasons of the logistics system in the COVID-19 pandemic. In this article, we have made an optimization on the basis of the FMEA method: the fuzzy is integrated into the FMEA algorithm, referred to as F-RPWN (fuzzy risk priority-weighted number). Meanwhile, the AHP is used to determine the weights of risk indicators. In this article, we consider new logistics failures, such as the failure modes and failure reasons of the logistics system under the COVID-19 pandemic. There are 12 failures that have been determined, and relevant preventive and corrective measures have been recommended to cut off the path of failure propagation and reduce the impact of failures. In addition, the proposed method can help logistics firms, their supply chain partners, and customers with risk management issues during the COVID-19 pandemic.