Preheat the oven to 350°F (175°C). Grease a 9×9-inch baking pan and set it aside.
In a medium bowl, mix the chopped peaches with lemon juice, sugar, and cornstarch. Set aside to macerate while you prepare the crumble topping.
In a large bowl, combine the flour, oats, brown sugar, baking powder, and salt.
Add the cold, cubed butter to the dry ingredients. Use a pastry cutter or fork to blend the butter until the mixture resembles coarse crumbs.
Stir in the vanilla extract.
Press about 2/3 of the crumble mixture into the bottom of the prepared baking pan to form the crust.
Spoon the peach mixture over the crust and spread it evenly.
Sprinkle the remaining crumble mixture on top of the peaches.
Bake for 35-40 minutes, or until the top is golden and the filling is bubbling.
Allow the bars to cool for at least 15 minutes before slicing.