Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch cake pan.
Whisk together yogurt, sugar, oil, eggs, coffee, and vanilla extract in a bowl. Add flour, cocoa powder, baking powder, cinnamon, baking soda, salt, and nutmeg; whisk until air bubbles form in the batter. Pour batter into the prepared cake pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 50 to 60 minutes. Cool on a wire rack for 15 minutes. Run a table knife around the edges to loosen. Invert carefully onto a serving plate or cooling rack. Serve warm or at room temperature.
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