The answer to this is that Sam used the bark of the slippery elm tree as a sort of soap. This is not to say that he actually used it to make soap. Instead, he used the bark itself as soap. He said that this was a kind of bark that got soapy when it was rubbed.
This can be found at the beginning of the chapter entitled "I Find A Real Live Man." When the chapter starts, Sam is recounting the daily routine that he follows in the summer. He says that he would take a bath in the spring every morning. When he would do that, he would rub himself with the bark of the slippery elm.