Inverse Galois problem for simple Lie type groups Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$ (a lot of cases known, but wide open in general).

I'm interested in progress for families of simple Lie type groups other than
  $PSL_n(q)$.

Important note (edit). I'm looking for results for complete families, not individual groups.
I haven't been able to find almost any information, and that suggests that the answer is "nothing is known". But it would be nice to have a reference for that, if it is the case.
On a side note, as an example of progress for those groups (so that they are not completely intractable), Belyi proved that the 6 families of classical simple Lie groups are realizable over $\mathbb{Q}^{ab}$.
 A: There is Thompson and Volklein, who prove that the symplectic groups are Galois groups:
Thompson, J. G.(1-FL); Völklein, H.(1-FL)
Symplectic groups as Galois groups. 
J. Group Theory 1 (1998), no. 1, 1–58. 
12F12 

More information on realization of simple groups can be found in Volklein's book:
MR1405612 (98b:12003) Reviewed 
Völklein, Helmut(1-FL)
Groups as Galois groups. (English summary) 
An introduction. Cambridge Studies in Advanced Mathematics, 53. 
Cambridge University Press, Cambridge, 1996. xviii+248 pp. ISBN: 0-521-56280-5 
12F12 

Finally, a zoo of low-order (not that low) is discussed by David Zywina.
A: It's hard to keep track of all relevant literature on the inverse Galois problem for finite groups of Lie type, but at this point many special cases have been worked out while others remain open.   One rather recent paper here indicates how subtle the approaches have become.   (For an arXiv preprint, see here.)
Work by Gunter Malle, including his joint book with B.H. Matzat, is an important source; Malle might also have advice to give by email.    (One obvious question is what kind of search tools you are using and what kind of library access you have?)
As you realize, finite special linear groups and their relatives pose extra problems---especially in low ranks.   The study of finite simple or almost-simple groups is natural but has so far been successful only in special cases rather than through some unified argument.
