The simple answer is...YES! Two things are necessary. First the batteries need to be connected (so that 12v works in your coach lights.) Second, your inverter needs to be turned on. There is a small remote panel, often by the door with a single push button to turn on the inverter. Once on, a green light will glow. Push again, and it will go off.
When on the road, your alternator supplies 12V to the batteries and the inverter takes 12V from that batteries and makes 110V for the frig. When you plug into shore power, there is an automatic transfer switch that will immediately feed the frig from the shore power and not the inverter, but the inverter will remain on but have no load os it uses very little power and can be left on as long as you are connected to shore power or are using the coach.
If your inverter is on but the frig isn't, there is a GFI receptacle on the inverter that may have popped so the frig won't get power when disconnected from shore. Just push the reset button...and all will be cool!