The first answer is not a database of the Steiner triple systems, but rather how many there are up to isomorphism.  [It is known][1] that they there are 2 of order 13, 80 of order 15, and 11084874829 of order 19.  The last number was computed by Kaski and Ostergard, and I suppose that the best approach is to ask them for their data.  It's more or less the end of the story, because it is easy to make larger Steiner triple systems, but impossible to compile  them into a complete database.  It looks like a number of people have the Steiner triple systems of order 15, but I didn't find a paper that simply lists them.

Actually [this paper][2] describes a compressed 39-gigabyte file with the Steiner triple systems of order 19, and says that it is available by e-mail request from three of the authors (including Kaski and Ostergard).

  [1]: http://www.research.att.com/~njas/sequences/A030129
  [2]: http://www.nbl.fi/~nbl4075/sts19cat.pdf