Preheat the oven to 350°F. Line an 8x8 baking dish with parchment paper and set aside.
In a medium mixing bowl, stir together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a separate large mixing bowl, whisk together the cane sugar, brown sugar, oil, apple cider, vinegar, and vanilla until smooth.
Pour the dry ingredients into the wet ingredients and stir just until no visible white flour remains. A few lumps are fine and expected.
Stir in the diced apples until evenly distributed.
Pour the cake batter evenly into the prepared baking dish.
Bake for 40 to 45 minutes until the top is golden and a toothpick inserted into the center comes out clean.
Transfer the pan to a cooling rack for 10 to 15 minutes then remove the cake from the pan and place it back on the cooling rack to continue cooling completely.
Once the cake has cooled, slice and enjoy!