I have been working recently on a similar problem, namely, maximizing the absolute value of the determinant of a chosen $m\times m$ submatrix $S$ of a $m\times N$ submatrix $V$ (think to the columns of $V$ as your vectors). At first sight, it may seem that the two problems are not related, but one can prove that if $S$ has this property then $S^{-1}V$ has all entries bounded in modulus by 1. This is enough to ensure some form of stability, at least in the applications that we were [investigating][3], and I suspect that yours case might be similar. Other useful references are [this paper][4] and [this one](https://math.berkeley.edu/~mgu/MA273/Strong_RRQR.pdf).

I can tell you the following.

* Finding the maximum volume submatrix is an [NP-hard problem][1], so I guess that your problem may suffer the same fate
* a useful relaxation, however, is finding a submatrix that has *locally* maximum volume, i.e., larger than all those that can be obtained by changing one vector only. There is a [paper][2] by Knuth (yes, *that* Knuth) that studies the problem. 
* a second relaxation is finding a submatrix $S$ such that $S^{-1}V$ has all entries bounded by some real $\tau>1$. This problem can be solved explicitly in $O(Nm^2\frac{\log m}{\log\tau})$. The good conditioning properties carry over to this relaxation, up to a factor $\tau$. Putting everything together, I think that one can prove using the techniques in our paper that the matrix chosen by the algorithm has conditioning which differs by at most a factor $O(\sqrt{Nm}\tau)$ from the conditioning of the rectangular matrix $V$ (defined as largest over smallest singular value), which is a theoretical maximum for the conditioning of $S$ (I think).
* if you have a practical computation and you want to check how things work with this solution, I have some [Matlab code][5] in a packaged and usable state that you might wish to try out. Feel free to ask for explanation if the documentation is too poor.


  [1]: http://dx.doi.org/10.1007/s00453-011-9582-6
  [2]: http://dx.doi.org/10.1080/03081088508817636
  [3]: http://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/906
  [4]: http://spring.inm.ras.ru/osel/?p=13
  [5]: https://bitbucket.org/fph/pgdoubling/overview