Preheat oven to 350°F Butter bundt pan well and dust with some flour, knocking out excess.
Sift together 2 1/2 cups flour, baking soda, baking powder, and salt into a bowl. Beat together butter and brown sugar in a large bowl with an electric mixer at high speed until pale and fluffy, about 3 minutes, then beat in vanilla.
Reduce speed to medium and add eggs 1 at a time, beating well after each addition and scraping down side of bowl occasionally, then beat until very smooth and fluffy, about 2 minutes more.
Reduce speed to low and add all but 1/2 cup flour mixture, mixing until just combined.
Toss zucchini, chocolate chips, and walnuts with remaining 1/2 cup flour mixture and add to batter, then mix batter with a rubber spatula (batter will be thick).
Spoon batter into bundt pan, smoothing top. Bake in middle of oven, rotating pan halfway through baking, until side begins to pull away from pan and a tester comes out clean, 45 to 50 minutes total.
Cool cake in pan on a rack 30 minutes, then run a thin knife around outer and inner edges. Invert rack over pan, then invert cake onto rack. Cool completely.