Attach the data lines (see here) to either the 'scope or probe so you can read the signals being sent. Then plug the modified cable into your computer, start up the program used to control the nerf gun, and finally record the data pulses associated with each specific command sent by the program.
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.
issue the commands on Windows (or whatever OS you use)
This is the part that is an unknown. If there is a utility/application/API that can be run on a PC to fire the gun, and it is feasible to include a PC in the solution, then all you need to do is send a command over the serial link to execute the command to fire the gun - no USB emulation needed, just use the standard Arduino USB drivers. Is such an interface available, though?