Preheat the oven to 375 degrees F. Mix together the flour, sugar, baking powder, apple pie spice, baking soda, and salt.
In a food processor, pulse the yogurt, carrots, oil, vanilla, and eggs. Mix the wet ingredients into the flour mixture, then stir in the raisins.
Divide the batter among a greased or paper-cup-lined 12-cup muffin tin. Bake until golden-brown, 25 to 28 minutes.