Wireless monitoring could greatly impact the fields of structural health assessment and infrastructure asset management. A common problem to be tackled in wireless networks is the electric power supply, which is typically provided by batteries replaced periodically. A promising remedy for this issue would be to harvest ambient energy. Within this framework, the present paper proposes to harvest ambient-induced vibrations of bridge structures using a new class of piezoelectric textiles. The considered case study is an existing cable-stayed bridge located in Italy along a high-speed road that connects Rome and Naples, for which a recent monitoring campaign has allowed to record the dynamic responses of deck and cables. Vibration measurements have been first elaborated to provide a comprehensive dynamic assessment of this infrastructure. In order to enhance the electric energy that can be converted from ambient vibrations, the considered energy harvester exploits a power generator built using arrays of electrospun piezoelectric nanofibers. A finite element analysis is performed to demonstrate that such power generator is able to provide higher energy levels from recorded dynamic loading time histories than a standard piezoelectric energy harvester. Its feasibility for bridge health monitoring applications is finally discussed.