There is a run/shut-dn fuel shut-off solenoid which performs this duty. The solenoid on MY 2014 RTV-X900 shuts off the fuel when the key is turned off…..and about ten seconds afterwards….you can hear a “click” which is when the solenoid relaxes back to the fuel “ON” position.
So…sounds like you need to confirm power is being directed to your solenoid when the ”ignition” switch is turned OFF…. and if-so…then see if the small actuator which closes the fuel valve is working or not.
There is a 15A fuse (in position 15) that is the shut-dn timer. There is a 5A fuse (ironically, in position 5) that is also a key-stop timer. So check fuses, of course. (and just to make things more-interesting….the RTV uses a “Micro-Mini” type of fuse….smaller than what is commonly found in autos these days.