This is a commonly asked question:

It is completely legal to shoot rabbits with a catapult with the intention to kill. So if you are looking for dinner you can use a catapult, it is legal.

However you must :
a. Have permission from the landowner to do so first!
b. Be proficient enough to be able to kill, and must use suitable projectiles to ensure this! For example steel ball bearings.
c. Intend to kill.

It is also important to note : The Wild Mammals Protection Act 1996 which makes it an offence to injure, maime, beat with sticks, torment, burn, wild animals etc. etc. In other words not to kill but merely to cause suffering. Therefore if you are seen shooting at a rabbit time and time again without killing or stunning the rabbit only hitting it, this could be seen as contrary to this act, and would need to answer for it, possibly in court. Therefore power, accuracy, using adequate and suitable projectiles, and at a range whereby an accurate shot will kill is vital to be, and be seen to be within the law.

A sling can be powerful enough to hunt game such as small rodents and birds at ranges up to 25 meters.A typical heavy pull band slingshot should be used with 9 mm (3/8") to 12 mm (1/2") steel balls. Using lighter ammunition doesn't increase the speed of the projectile significantly. In the UK where there is a strong following of hunting with catapults, a variety of game is taken amongst them hares, rabbits, pheasants, geese, ducks, pigeon, dove, rook, and all manner of other species. Usually expert catapult hunters use lead ball projectiles often sized at .44 and .50 calibre which mostly they cast themselves using musket ball moulds.