Computing volume of high-dimensional convex polyhedra
You can try and use qhull http://www.qhull.org/ - it can compute the volume of the convex hull of the vertices. However, a priori I do not see any reason for it to perform reasonably for your range of numbers. If qhull does not work, you can try CGAL/GALIA. In the worst case, you can try and impelement one of the random walk algorithms you mentioned - they should not be too hard to implement in this case, but the constants involved might be very large...