How widespread are electronic-only dissertations?
The first page of a Google search shows UVA and USC do this already.  You could try contacting colleagues at those schools and see how the experience has been.  (Sorry, I don't have a statistic.)

P.s. I am sympathetic to those who prefer the bound volumes.  I think you can in some ways scan a text and multiple texts more easily in the hard copy than electronic.  Also, there are excellent arguments that books on acid free paper are a superior storage media than computer files (longer lasting, safer, future readable).  Finally I think that the granting university is more likely to have future students working in the same general area.  Thus they benefit especially from a printout.