When it comes down to options of staying at a Ryokan, there are many options on the Izu Peninsula, let alone, the town of Higashiizu.
Here are some of the places I have had affiliations with, so it is not necessarily recommendations based on preferences and likings but rather the ones I have visited. (Either a friend or an affiliate had a chance to stay at one of these accommodations)
熱川荘 starting from 6480yen per person http://www.ataso1.yad.jp/index.html
– a very “showa” taste, humble accommodation. Atagawa is filled with public onsen so you can buy a seperate ticket for ¥1,200 (3 tickets) and you can visit any of one of the 17 onsens located in the nearby hotels. http://www.atagawa.net/yumeguri.html
いさり火 starting from 31400yen per room http://www.isaribi.co.jp/eng/index.html
-private rooms with nice view in a very quite setting.
銀水荘 starting from around 10,000 yen per person http://www.inatori-ginsuiso.jp/
– national award winning ryokan-style hotel. Has gorgeous baths that fit many people.
If you type in specific areas such as “Inatori” and “Atagawa” you will have a variety of recommendations.
Our town name is called “Higashiizucho (or just HIgashiizu) ”.
If you can put this in the search, you will most likely find an accommodation too!
Hope this helps. Please ask if you have any more questions!