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I have students creating a basketball launcher device using ping pong balls and it is controlled using an Arduino Decimila board. They have test all the parts separately and they work fine. The device must launch a ball at one of three targets. A stepper motor controlled by 4 TIP31 transistors is used to control the rotor to make this possible. A electromagnet is used to draw a swingarm backwards. The motor to pull the swingarm back is a servo motor. If the stepper is not activated, the servo works fine. If the stepper is run and then the servo, it loses power somehow and seems to lose its mind, behaving erratically.

I suspect some sort of electromagnetic interference or trouble with the common grounds between the Arduino 5V source and external power supply that provides 9V to the electromagnet. It sure is frustrating though. Any words of advice?

You lost me after the first sentence... A basketball launcher device using ping pong balls? Stepper motor driving what rotor, an electromagnet draws the swing arm back but the motor to pull the swingarm back is a servo?
Post schematics and perhaps pictures of the physical setup. Currently you're not making enough sense aside from EMF from the motors to the Arduino to warrant a response.