Actually, a very similar statement can be found in the paper [Reflexive pull-backs and base extension][1] by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S_2$ and the sheaf does not need to be a line bundle only coherent and flat over the base. We develop a little bit of a relative theory in section 3, so you might find the rest useful as well, especially the statement regarding a characterization via local cohomology. We did assume the codimension two condition for every fiber, but it might not be necessary actually. I would have to check the proof (I will try to do that sometime). I would add that (as you discovered) for a Hartogs type statement you do not need normality and many things can be done for $S_2$ schemes that are usually done for normal ones. The main reason normality is more widely used (besides that it is easier to define and *think* that one understands it better) is that working with divisors on $S_2$ but not normal schemes has to be done very carefully. See [this][4] MO answer for more on this and perhaps  [this][2] and [this][3] for more on Hartogs type questions.


  [1]: http://www.ams.org/journals/jag/2004-13-02/S1056-3911-03-00331-X/
  [2]: https://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-hartogs-phenomenon/45354#45354
  [3]: https://mathoverflow.net/questions/45347/why-does-the-s2-property-of-a-ring-correspond-to-the-hartogs-phenomenon/45616#45616
  [4]: https://mathoverflow.net/questions/35736/the-canonical-line-bundle-of-a-normal-variety/46663#46663