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.

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).

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.  

**Update 1:** It seems that everyone works from the paper "Small Steiner triple systems and their properties", by Mathon, Phelps, and Rosa.  This paper is basically an encyclopedia of the 80 Steiner triple systems of order 15 and many of their properties.  It also introduces a somewhat standard numbering.  The thing to do at this point would be to transcribe the data in this widely cited paper into a file.  Google seems to indicate that no such file has been posted to the web.

**Update 2:** It was done!  See [this GAP file][3] by Nagy and Vojtechovsky.  For some reason, a Steiner triple system of order $n$ is also called a Steiner loop of order $n+1$, and that is the terminology that they use.  They copied the data from Colbourn and Rosa, Triple Systems, which presumably is the same as in Mathon, Phelps, and Rosa.

How I found it:  I Googled one of the hexadecimal strings used to describe one of the STS(15)s.  None of the Google's heuristics worked for me, so instead I used the old-fashioned trick of searching for a very specific keyword.  It also shows up in a LaTeX PHD thesis, but the GAP file, which has almost the same syntax as Python, is better.

  [1]: http://www.research.att.com/~njas/sequences/A030129
  [2]: http://www.nbl.fi/~nbl4075/sts19cat.pdf
  [3]: http://www-groups.dcs.st-and.ac.uk/gap/Manuals/pkg/loops/data/steiner.tbl