I don't remember but probably. Looking at the pictures it looks like it has a thermal relay too.
It is most likely a problem with your light on/off switch. To test if that is the problem check resistance with a multimeter between the two points on the switch. If the resistance doesn’t change when you switch the button then it is a faulty switch. If you don’t have a multimeter you can physically remove the wire and verify the light goes off.
I am 99% sure that you will never find that part. any of these components aren't really meant to be replaced. Things like the battery or the ac-dc converter you can find because they just used off-the-shelf parts. You could probably find that size hose but you would have to find a way to seal/crimp it onto the air nozzle on the other end.
To charge the battery? The back of the unit will say what the "input" should be.