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 $\mathbb{Q}^{ab}$.