PeterH:
Using USB devices is easy, but writing USB device drivers is hard. Presumably you could interface to the NERF gun mechanically or electrically rather than electronically, if you wanted. That would eliminate the hardest part of the problem.
Another approach using a Leonardo is have it simulate a keyboard and/or mouse, and have it connected to the PC with the Nerf attached via USB and issue the commands on Windows (or whatever OS you use) to fire the Nerf gun. You can probably do this on an Uno with a USB host shield.