You absolutely can use your washer/dryer while not connected to city water.
They are designed as "efficient" washers that use very little water, (about 5 gallons per load). If your have 80-100 gallon fresh water tanks, that's a lot of loads.
The combo washer/dryer units actually use more water per load. They use cold water to condense steam during the dry cycle, so they use more water.
If you are in a motorhome that was factory plumbed for a washer/dryer, you very likely are carrying plenty of water for a load or two without any concern.