Whisk flour, baking powder, baking soda, and salt together in bowl. Whisk sugar, melted oil, water, almond butter, and vanilla in large bowl until well combined and smooth. Using rubber spatula, stir flour mixture into oil mixture until just combined; fold in chocolate chips.
Cover bowl with plastic wrap and let rest at room temperature for at least 1 hour or up to 4 hours. (Dough can be refrigerated for up to 24 hours; let sit at room temperature for 30 minutes before portioning.)
Adjust oven rack to middle position and heat oven to 350 degrees. Line 2 rimmed baking sheets with parchment paper. Divide dough into 16 portions, each about 3 tablespoons, then arrange dough mounds 2 inches apart on prepared sheets.
Bake, 1 sheet at a time, until light golden and edges have begun to set but centers are still soft, 12 to 14 minutes, rotating sheet halfway through baking. Let cookies cool completely on sheet. Serve. (Cookies can be stored at room temperature for up to 3 days.)