Preheat the oven to 375°F (190°C). Line a 12-cup muffin tin with paper liners or grease the cups.
In a large bowl, combine the rolled oats and Greek yogurt. Stir well and let sit for 5 minutes to soften the oats.
In a separate medium bowl, whisk together the sugar, vegetable oil, and eggs until smooth.
Add the wet mixture to the oat and yogurt mixture and stir until combined.
In another bowl, sift together the all-purpose flour, baking powder, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Stir in the vanilla extract until evenly distributed.
Spoon the batter evenly into the prepared muffin tin, filling each cup about 3/4 full.
Bake for 18-20 minutes, or until a toothpick inserted into the center of a muffin comes out clean.
Remove the muffins from the oven and allow them to cool in the tin for 5 minutes before transferring to a wire rack to cool completely.