In this study, a green and sustainable hydrometallurgical process, based on soft acid derived from agrifood waste as a byproduct, is used to recover molybdenum from selective spent oxidation catalysts for formaldehyde production. Tartaric acid recovered from winery waste is used as a leaching and chelating agent. The spent catalyst was characterized by X-ray diffraction, scanning electron microscopy, and inductively coupled plasma optical emission spectroscopy. The last term was also used to study the leaching efficiency. Under optimized conditions, a molybdenum recovery of 87.36% ± 2.94 wt % was achieved: 1.3 M tartaric acid, 75 g/L solid−liquid ratio, and 60 min at 25 °C. Moreover, the leaching kinetics were also investigated using the shrinking core model, which is correlated to each step of the leaching process, including chemical reaction, product layer diffusion, and film diffusion control. The step that exhibits the best agreement with the experimental kinetic data is considered as the rate-controlling step. The proposed hydrometallurgical process was found to be simple, efficient, and environmentally friendly. Using agrifood wastes, it becomes possible to process industrial waste to recover and reintegrate expensive metals for an efficient circular economy.