Preheat oven to 325 degrees. In a large bowl, mix butter and sugar with a wooden spoon until pale and fluffy. Stir in vanilla, lemon zest, and salt. Add flour and cornmeal and mix until mixture is crumbly. Add zucchini and stir until a thick dough forms.
Drop dough by rounded tablespoons, 2 inches apart, onto two parchment-lined baking sheets. Bake until cookies are light golden brown at edges, 25 to 30 minutes, rotating sheets halfway through. Let cool completely on wire racks.