co.combinatorics - Finding integer points on an N-d convex hull

Because the facets of your convex hull are themselves polytopes (of one lower dimension—$d{=}21$ in your case), it seems your question is equivalent to asking how to count lattice points in a polytope. One paper on this topic is "The Many Aspects of Counting Lattice Points in Polytopes" by J. A. DeLoera. Section 4 is entitled, "Actually Counting: Modern Algorithms and Software." A key reference in that section is to a paper by Barvinok entitled, "Polynomial time algorithm for counting integral points in
polyhedra when the dimension is fixed" (Math of Operations Research, Vol. 19, 769–779, 1994), whose title (polynomial time) seems to provide an answer of sorts to your question.