While you are getting those links that PaulS mentioned... 
Theoretically it would be possible, but you'd have to know the command signals sent over the USB's data line. If they aren't in the documentation, you'd need an oscilloscope or logic probe and a little work to determine the necessary commands. Basically you'd take USB (use a 2.0/1.1 cable, I'm sure this type of desktop toy doesn't need a 3.0 USB bus) with at least one USB plug A and cut that plug off along with some spare cable. 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.