Building an electrically controlled spring based catapult
You could probably do the whole thing with one motor and a solenoid using the same principle as a car's start motor / starter solenoid.  Basically, mount a gear to the main arm.  The solenoid would push the drive gear (which is attached to the motor with a worm gear) against the main arm gear to lower it down.  To launch, disengage the solenoid.  

Note that you can make your own simple solenoid with a simple electromagnet / spring in opposition.

I haven't thought a whole lot about the loading mechanism... Some more information on the size of the apparatus would be helpful.