Ambient vibration energy is highly irregular in force and frequency. Triboelectric nanogenerators (TENG) can convert ambient mechanical energy into useable electricity. In order to effectively convert irregular ambient vibrations into electricity, the TENG should be capable of reliably continuous operation despite variability in input forces and frequencies. In this study, we propose a tandem triboelectric nanogenerator with cascade impact structure (CIT-TENG) for continuously scavenging input vibrations with broadband frequencies. Based on resonance theory, four TENGs were explicitly designed to operate in tandem and cover a targeted frequency range of 0-40 Hz. However, due to the cascade impact structure of CIT-TENG, each TENG could produce output even under non-resonant conditions. We systematically studied the cascade impact dynamics of the CIT-TENG using finite element simulations and experiments to show how it enables continuous scavenging from 0-40 Hz even under low input accelerations of 0.2 G-0.5 G m/s2. Finally, we demonstrated that the CIT-TENG could not only scavenge broadband vibrations from a single source such as a car dashboard, but it could also scavenge very low frequency vibrations from water waves and very high frequency vibrations from air compressor machines. Thus, we showed that the CIT-TENG can be used in multiple applications without any need for redesign validating its use as an omnipotent vibration energy scavenger.