You might want to study the work of Detinko, Flannery, and co-authors. For example:
Detinko, A.; Flannery, D. L.; Hulpke, A., Zariski density and computing in arithmetic groups, ZBL06825254.
Detinko, A. S.; Flannery, D. L.; Hulpke, A., Algorithms for arithmetic groups with the congruence subgroup property., J. Algebra 421, 234-259 (2015). ZBL1319.20040.
(moving comment up to the main body): They find things like the congruence depth (the index of the principal congruence subgroup), so the co-volume, and then (at least in the special linear and symplectic case you can use the Minkowski model (the PSD cone) of the symmetric space to construct the fundamental polyhedron (the bound tells you when to stop), so pretty much what you are asking for.
In the real hyperbolic case, there is the very interesting paper of Mark and Paupert:
Presentations for cusped arithmetic hyperbolic lattices, by Alice Mark and Julien Paupert.