Well, polynomial rings are very complicated. I assume the question is more generally about $GL_n(\mathbb{F}_q[T_1,\dots,T_m])$. 

The case of one variable can be deduced from the following paper:

- C. Soulé: Chevalley groups over polynomial rings. In: Homological group theory (Proc. Sympos. Durham) London Math. Soc. Lecture Notes Series 36 (1979), pp. 359-367. 

In this, Soulé computes the fundamental domain for the action of $GL_n(k[T])$ on the associated Bruhat-Tits building. Any finite subgroup will have a fixed point on the building, so it will appear as subgroup of a stabilizer of the $GL_n(\mathbb{F}_q[T])$-action on the building. The stabilizers are explicitly determined, they are semidirect products $U\rtimes L$ where $L$ is a parabolic subgroup of $GL_n(\mathbb{F}_q)$ and $U$ is a subgroup of the upper-triangular matrices with $\mathbb{F}_q[T]$-coefficients (in particular the latter is a $p$-group). Probably everything you want to know about finite subgroups and their conjugacy classification can be deduced from that. 

In the case $m>1$, several variables, a lot less is known. One can still use Soulé's theorem, applied to $GL_n(\mathbb{F}_q(T_1,\dots,T_{m-1})[T_m])$. Any finite subgroup will have a fixed point on the building and therefore be conjugate to a subgroup of one of the standard stabilizers described above (but of course for the base field $k=\mathbb{F}_q(T_1,\dots,T_{m-1})$). However, the conjugacy involved will be by matrices in $GL_n(\mathbb{F}_q(T_1,\dots,T_{m-1})[T_m])$ which may not be what you are interested in. Also, there may be some finite subgroups which only appear after some denominators are introduced. Anyway, this confirms the description in YCor's comment.

For $m>1$, I don't know of a comprehensible description of a space on which $GL_n(\mathbb{F}_q[T_1,\dots,T_m])$ would act with finite stabilizers with identifiable fundamental domain. This is already terribly complicated for $GL_2(\mathbb{F}_q[T_1,T_2])$, see 

- S. Krstić and J. McCool. Free quotients of $SL_2(R[X])$. Proc. Amer. Math. Soc. 125 (1997), 1585-1588. [(paper can be found here)][1]

A consequence of their result is that $SL_2(\mathbb{F}_q[T_1,T_2])$ surjects onto a free group of infinite rank. This provides lots of elements of infinite order and probably 
also tells us that the conjugacy classification of finite subgroups is very complicated. 
  [1]: http://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03809-4/