I did a solo kayak-self support trip down the Owyhee last year (April 26 to May 1) and really enjoyed it. I'm hoping to do another run again this year and have been keeping an eye on the snow pack, etc. Out of curiosity, I started Googling commercial trips to see when they run it and how often. This was by no means an exhaustive search, but I came up with seven outfitters running 16 different trips between 4/12 and 5/13. Almost half the trips were in the last week of April.
As permits become harder and harder to snag on the 4 Rivers, I think the Owyhee will see increasing pressure. At what point usage of the Owyhee increases to the point that permits become limited, I don't know. Right now, it still feels like a "remote" run, but it's really not much of a drive from Boise. Campsites seemed to be in fairly good shape last year, but that was after several years when the runoff was so meager there were few private or commercial trips.