Did/do they have access to moss at all times during the year? Did they wash the seal skins or just throw them out? Just rinse in cold water?
They either would have stored or found a substitute during moss lean times.
These days they* have MCN's (modern cloth nappies) the idea is that you line it with a disposable cloth and then use a reusable cloth for the bulk of the nappy. It's supposed to reduce landfill waste (that's another story) how ever the Inuit's, and others, in all likelihood used the same method. So washing would only occur periodically or if the skin was soiled. The washing of a soiled hide would be easier and faster (in both washing and drying time) than the cloth we use today.
*'they' being western contemporary women.