What provides the acid for leavening in this banana bread recipe?
Most of the ingredients in this recipe are acidic. 

ph values:


flour 5.5-6.5
brown sugar slightly acidic
bananas 4.5-5.2
butter 6.1-6.4


The salt is neutral and the eggs are not acidic, ph 7.1-7.9

The combination of these ingredients is acidic enough to interact with the baking soda to leaven the bread. 

Even without the bananas, there would be some leavening. Consider the standard ingredients in chocolate chip cookies: flour, sugar, butter, eggs, salt, baking soda and chocolate chips. Though they don't raise nearly as much as banana bread, they do poof up a bit.