Update:
The following is a very long-winded account of my experience booking Odoriko and Saphir Odoriko tickets (from Tokyo to Kawazu) using a JR green pass, which turned out to be somewhat complicated. If somebody is thinking of doing this, they might possibly want to wade through my account; otherwise there is absolutely no point. It would be faster and easier just to call JR and ask them how to handle things. And there might very well be a better way to do it than I ended up with, especially if you can deal directly with Izu Kyuko Railway to make advance reservations and pay for them (I did not do this). It will also much easier if you donft try to book the trains using a nationwide JR green pass. If you have one of the JR East regional passes mentioned on the following Japan Guide page, it is probably as easy as pie to book the gplainh Odoriko (assuming seats are available).
https://www.japan-guide.com/e/e6305.html I went to the Kawazu festival on Sunday, February 26, and rode the Odoriko and the Saphir Odoriko trains (departing from Kawasaki and returning to Yokohama). It turned out to be peak bloom time and perfect weather, which brought out very large crowds, making the train bookings fuller than they would ordinarily be. So my partner and I were extremely lucky to get seats using our JR green passes. As it turned out, JR would only reserve the seats for us as far as Ito and back. If (like the poster Bill above) you are purchasing tickets (i.e., not using a pass), then you can do so for the full length that the Odoriko and Saphir Odoriko trains run, including the Izu Kyuko sections. But for JR (green) pass holders, JR will only issue tickets (at no cost) as far as Ito. Then you have to deal with Izu Kyuko directly to get reservations and tickets for the segment beyond Ito and the return to Ito. The agent at the JR Travel Service Center in Kawasaki, where my partner and I redeemed our rail pass vouchers and got the tickets to Ito, said that normally the Odoriko is not sold out below Ito and we could just stay on the train, probably in the same seats, and arrange the booking of them with the Izu Kyuko conductor on the train. The JR agent said we probably wouldnft have any problem getting the Saphir back to Ito because people normally donft book it only for that segment. This worked out fine as far as getting to Kawazu went. The Izu Kyuko conductor came by after we left Ito and he said we could stay in our existing seats unless someone with a ticket for them came, and if that happened we could take any available seats. Then we needed to settle the costs when we got out in Kawazu, at a ticket adjustment window inside the exit wicket. And we would need to book seats (if possible) on the Kawazu to Ito segment once we got to Kawazu. Indeed, we kept our seats to Kawazu and paid the Izu kyuko charge when we got out. I think it was something like 2400 yen (we were in the green car of the Odoriko). So that part was fine, but the Saphir we had tickets for (from Ito to Yokohama) was sold out from Kawazu to Ito. Fortunately, there was an Odoriko that left a little earlier that had seats available, and we were able to book those (for something like 1875 yen as I recall; they were not green seats but were very nice ocean-side seats). So we had to leave Kawazu a little earlier than planned and wait about 30 minutes in Ito for our Saphir to show up. The timing was actually fine, as we had allowed ourselves enough time in Kawazu that cutting it a little short didnft really matter.
So anyway, as far as the train reservations and fees are concerned, using a JR green pass and settling with Izu Kyuko for their segments, it can be tricky. In the worst case scenario, you could take local trains on the Izu Kyuko line for those segments (and stand if necessary), but those trains are quite slow, so for something like a day trip, that fallback solution might be untenable if you wanted to do anything other than just ride trains up and down the Izu peninsula. On a weekday in the off season, it would be a much less risky proposition, I think.