Nah, you just put a 5-10kwh battery on there, along with a small water tank and pump - a load of washing (in Europe) only uses 50L and about 2kwh tops - up to 5 with drying. The rest of the battery powers the drive train for the vehicle. Then you just have a dock for charging and water replenishment. We only use zero phosphate eco detergent, so waste can be dumped wherever. Maybe position it to water some plants when it drains.
You know, my whole life is madcap projects (today is running half a km of fibre down the inside of a live water main) so I’ll update you when I have a mobile laundromat.