If I were in the position without a washing machine I would buy a used one. The last one I owned came from an appliance repair guy. He said that when many people upgrade their machines they needed him to haul away the older ones. He sold a bare bones one year old washing machine to me for just $150.
You could put the machine on a cart with wheels and connect it to your kitchen sink faucet. The water in the machine could drain into the sink. There are rubber adapters that fit over a faucet. They're like socks. The other end screws into the intake hose. You wouldn't need to connect both the hot and cold sides. Just use the hot side for a warm rinse cycle or in the middle of the cycle turn off the hot water and turn on the cold water.
Over time you would save money with your own machine.
Plungers with buckets do save money but they take up your time. I would use that method if I were in an RV away from a city. Kookaburra1701 let us know how the plunger works. If it works well I might get one to have as a backup machine. I've already got a big bucket.