Is it possible to run two small motors off an Arduino and have the motors move separately based on a random number generator?
Leaving your mechanical equipment uncommented *), an Arduino itself cannot drive 7.4V 1A Stepper Motors. An Arduino itself cannot drive any motor. You need a sufficient power supply and motor driver electronics. And there's a vast amount of different devices to choose from.

But then, the answer to "is it possible (random, two motors, moving independently)" is yes.

*) Well, regarding "uncommented" :
200 lbs is quite heavy to simulate/produce a few raindrops. Small motors won't move such gear as in your pic.