Preheat the oven to 350 degrees F (175 degrees C). Grease 18 muffin cups or line cups with paper liners.
Whisk 2 ½ cups flour, ½ cup oats, pumpkin pie spice, baking soda, baking powder, and salt together in a bowl.
Whisk pumpkin purée, 1 cup brown sugar, white sugar, vegetable oil, eggs, applesauce, and vanilla extract together in a separate bowl; stir in flour mixture to combine. Fold in raisins.
Beat ¼ cup brown sugar and butter together in a bowl until smooth and creamy; whisk in 2 tablespoons oats and 2 tablespoons flour using a fork until streusel topping is crumbly.
Divide batter among the prepared muffin cups, filling each ⅔ full; sprinkle with streusel topping.
Bake in the preheated oven until a toothpick inserted into centers comes out clean, 25 to 35 minutes.
