A volume integral (VI) equation method based on the fast Fourier transform (FFT) is proposed for the electromagnetic simulation of complex systems in large magnetic confinement fusion (MCF) devices. Such FFT-VI allows for dramatically reducing the time and the human effort required to adapt the CAD input files to models suited for electromagnetic simulations. Moreover, with FFT-VI the mesh generation is performed automatically and is very efficient and robust. The adoption of the FFT allows for significantly reducing the computation time and the memory requirements, therefore making simulations with several millions of unknowns feasible. Numerical results show that, in terms of numerical performances, the FFT-VI method applied to challenging electromagnetic problems in representative MCF devices is markedly superior with respect to the state of the art VI techniques. A sample implementation of FFT-VI is made publicly available.