Heat the oil in a large nonstick skillet over medium heat. Add the onion, mushrooms, and chorizo and cook, stirring and breaking up the chorizo with the back of a spoon, until brown and cooked through, 3 to 4 minutes. Transfer to a bowl and let cool to room temperature.
Working one at a time, place a wrapper on a clean surface. Place a rounded tablespoon (15 g) of the chorizo mixture and a little cheese in the center. Brush the outer edge of the wrapper lightly with water, then fold over the sides to meet each other and pinch together tightly to seal.
Transfer the empanadas to the Air Fryer rack and cook at 380°F (195°C) for 8 minutes, then flip and cook until deep golden brown, 2 to 3 minutes longer.