There isn't an option for joystick press/release in the input map.
The joy button list covers all the buttons except for the joysticks. The joy axis list has loads of options however except for 'Axis 6 + (L2)' & 'Axis 7 + (R2)' there's no information of the axes below Axis 3 + so could it possibly be any of them?
the input code for the joystick presses is
Input.is_joy_button_pressed(0, 8) #Left joystick
Input.is_joy_button_pressed(0, 9) #Right joystick
but would this work for all controllers?