Think of it this way. The new illuminated switch is basically the same as the old switch with an added LED. Once the switch is moved to the On position, the load is now hot but for the LED to be lit, it requires the ground to complete the circuit specific for that LED.
The ground for that new switch does not have to be an independent ground as you can tie into another ground nearby. Maybe there is a 12V light where you can use that ground? Also, if yours is like mine where there is a "reset" LED in the same switch plate, there is a ground there you can use. You will just need to test which is the ground wire.