A copper-based metal–organic framework (MOF-199) demonstrates excellent performance and considerable potential in the gas detection of volatile organic compounds (VOCs). However, the powder of MOF-199 seriously limits its application in the fabrication of flexible devices. Herein, we introduce a simple method of blending electrospinning, combining MOF-199 with polyacrylonitrile (PAN) nanofiber to form a MOF-199/PAN nanofiber membrane with the flexibility of PAN nanofiber and the functionality of MOF-199. The capacitive sensor comprising MOF-199/PAN nanofiber membrane and carbon nanotubes exhibits good flexibility, high porosity, and sufficient adsorption active sites. The sensor has the highest response to methanol among seven target VOCs. It exhibits outstanding sensitivity, long-term stability, good linear response, and repeatability toward methanol (250–1250 ppm). Owing to the flexibility of the MOF-199/PAN nanofiber membrane, the sensor maintains a stable sensing response after repeated bending at 180˚–0˚ 50 times and exhibits a certain degree of response in the 180˚ bending state. The results show that the sensor has flexibility potential. These provides some ideas and reference for the development of VOCs sensors.