We present a methodology to simulate the quantum thermodynamics of thermal machines which are built from an interacting working medium in contact with fermionic reservoirs at fixed temperature and chemical potential. Our method works at finite temperature, beyond linear response and weak system-reservoir coupling, and allows for non-quadratic interactions in the working medium. The method uses mesoscopic reservoirs, continuously damped towards thermal equilibrium, in order to represent continuum baths and a novel tensor network algorithm to simulate the steady-state thermodynamics. Using the example of a quantum-dot heat engine, we demonstrate that our technique replicates the well known Landauer-Büttiker theory for efficiency and power. We then go beyond the quadratic limit to demonstrate the capability of our method by simulating a three-site machine with non-quadratic interactions. Furthermore, we demonstrate the capability of our method to tackle complex many-body systems by extracting the super-diffusive exponent for high-temperature transport in the isotropic Heisenberg model.