The answer to your stated question is certainly "no", which I can support indirectly by reference to the simpler case of rank 1 groups.
The basic question here is natural but is already quite difficult even in rank 1: study the representation theory of a given group scheme over a ring of $p$-adic integers (or more generally, the ring of integers of a local field) by working out the representations of the finite groups over finite residue class rings .
This type of question has a fairly long history, so it may be worthwhile to follow the paper trail. There are for example a number of relevant papers in the rank 1 case, including one by Kutzko here and a little later by A. Nobs and J. Wolfart here and here.
It's important here to keep an open mind about techniques not encountered in the study of matrix groups over finite fields. That study was done in a special case in the Simpson-Frame work (which has some minor errors), building on the recursive combinatorial determination of characters of finite general linear groups by Green in 1955. There was also the work of his student Srinivasan on $Sp_4$ and Chang-Ree on groups of type $G_2$ over finite fields, and ultimately the far more sophistiated work that flowed out of the 1976 Deligne-Lusztig construction of virtual characters. By now the theory over finite fields has acquired considerable detail through the efforts of Lusztig and others. As far as I can tell, there is still no overall program for constructing the representations (or even the characters) of the finite groups of Lie type over rings such as $\mathbb{Z}/q \mathbb{Z}$ which are not fields.