You'd have to bring your own soap. A cup of powder would do multiple loads. The Appalachian shelters are maintained by volunteers, to a basic minimum. There's usually a broom and some ropes for hanging stuff up and I assume they do some repairs to make sure you don't fall through rotten floorboards at night. In addition to some rowdy weekend hikers there are a lot of through hikers who spend 6 months doing the whole trail and might have to walk several miles off they trail quite a few times times along the way to find a laundromat. A simple hand cranked agitator in a tub every hundred miles or so that you fill with a bucket from the creek and add your own tablespoon of powder sounds like it would save hundreds of hours of side trips and waiting for washer and dryer time, and tens or hundreds of dollars in cash for people who want to devote their energies to the trail itself. You would arrive at the shelter, spend 15 minutes or so washing and rinsing one load consisting of one pair of pants and two or three sets of socks, shirts and underwear, hang them overnight, and be on your way in the morning at virtually no cost in time or money.

The thing wouldn't be stolen as it would be too heavy for some idiot to bother carting out, so the worry would be wearing out, or vandalism.