Preheat the oven to 350°F (175°C), grease a 9x5-inch loaf pan, and lightly squeeze the grated zucchini so it is moist but not dripping.
Whisk the flour, baking soda, cinnamon, and salt together in a medium bowl until evenly combined.
In a larger bowl, whisk the eggs, brown sugar, vegetable oil, and vanilla until the mixture looks smooth and glossy.
Fold the grated zucchini into the wet mixture, then add the dry ingredients and stir gently until only a few flour streaks remain.
Fold in about three-quarters of the chocolate chips, taking care not to overwork the batter.
Spread the batter evenly into the prepared pan, scatter the remaining chocolate chips over the top, and bake for 50–55 minutes.
Let the bread cool in the pan for 15 minutes before transferring it to a wire rack, then wait until mostly cool before slicing.