Preheat the oven to 350 degrees F (175 degrees C). Grease a 12-cup muffin tin with cooking spray or line with paper liners.
Stir flour, oats, baking soda, baking powder, cinnamon, and salt together in a large bowl.
Beat eggs, applesauce, maple syrup, and vanilla in another bowl. Pour egg mixture into flour mixture and stir until evenly combined. Stir in carrot and zucchini. Spoon batter into the prepared muffin tins, filling each cup 2/3 full.
Bake in the preheated oven until a toothpick inserted into center comes out clean, 15 to 20 minutes. Cool in the muffin tin for 5 minutes. Transfer to a wire rack and let cool completely, about 20 minutes, before serving to hungry toddlers.
You can use rice flour in place of whole wheat flour, if preferred.
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