@few_reps: This is very interesting. Please let me know how it looks like. In Fuchs' paper, I found four generators for the Apollonian group and a geometric description of the corresponding arrangement of reflection planes, tangent to each other at infinity.

Actually (silly me again!), in his recent paper, John McLeod says that Vinberg's algorithm (as MacLeod applies it in arxiv.org/abs/1007.2299) produces an infinite-volume polytope in dimension $14$ and does not halt in dimensions $\geq 15$.

All right, that's silly me: what Vinberg says in his original paper is that if finding $m$ normals and the respective half-spaces $H^{-}_i$, $i=1,...,m$, we have that the volume of $P=\cap^m_{i=1} H^{-}_i$ is finite, then the algorithm halts at the $m$-th step.

I'm afraid here my ignorance comes into play more than yours: what is usually done by people using Vinberg's algorithm is that they search for a representation $O(q) = \Gamma\rtimes H$, with $H$ finite and $\Gamma$ generated by reflections. Then either they find a finite-volume polytope generating $\Gamma$ by reflections in its sides or $\Gamma$ is not finitely generated by reflections (so the algorithm does not stop, finding more and more normals for new "facets"). I will look into some papers for what you ask about and get back to you.

I'm sorry, I messed up everything: I think that here "reflective" means that the maximal subgroup of $O(q)$ generated by reflections is finite-index in $O(q)$. Indeed, the group $O(q)$, with $q$ as in your question, is generated by 10 reflections and 2 isometries of infinite order, and is not reflective (because of the homology argument above), according to few_reps. The Apollonian group is reflective but has infinite co-volume.

Well, my little wild speculation is really too wild there: the groups is non-reflective! Again.

If this group is non-reflective then Vinberg's algorithm should not stop (indeed, otherwise the group would be reflective though). The fundamental polygon has 10 reflective sides and 4 sides identified in pairs by the infinite-order isometries, in'nit? Sorry if I got it wrong. Actually, I would be happy with any description, not necessarily a picture (however, a picture could be very beautiful ...)

What kind of methods do you use? I just wonder if one has a kind of Vinberg's algorithm, producing nice polytopes/polyhedra. Can you visualise the fundamental domain (Poincare's polygon) for this action in the upper half-plane? What are the angles of the fundamental polygon?