At one point during the final chase scene, you see Hanna (
Saoirse Ronan) lean down and snatch a length of bungee cord from one of the rides. She uses this to launch the arrow. She most likely used a method known in the wilderness as 'elbowing'. The bungee and crease of the butt of the arrow align with the elbow and arrow tip and thumb with the thumb acting as a trigger. Other methods could have been employed though as the chase had gaps in visuals.