The Japan Knowledge database is now providing the electronic version of the Genji Monogari, with searchable text, from the Shinpen Nihon Koten Bungaku Zenshu (SNKBZ) edition (http://www.jkn21.com/top/corpdisplay).
This is the first work from the SNKBZ collection to be released in this new format. Two more works will be released very soon.
Japan Knowledge also provides access to the full-text collection of Toyo Bunko 東洋文庫 (in PDF format).
In order to search those collections separately, please go to Japan Knowledge and scroll down the home page until you find the “叢書・ＵＲＬ” section, at the right-hand side of the screen, and click the "個別" button.