Delaunay Triangulation on Convex Polytopes — Uniform Sampling
This is an active area of research. Fortunately, recently there has been a breakthrough due to Vempala and Cousins [1]. They provide a MATLAB implementation of their algorithm. If you're interested in the technical details, you can check out some slides from a recent workshop.

This algorithm only samples points approximately uniformly. Exact uniform sampling, or rather the related volume estimation problem, is known to be hard (under reasonable assumptions) in general. Your convex body isn't general, however – it's a polytope. Nevertheless, the answers to this question suggest that there is no better algorithm for this special case (they suggest employing general algorithms which are older than Cousins–Vempala).




Bypassing KLS: Gaussian Cooling and an $O^*(n^3)$ Volume Algorithm by B. Cousins and S. Vempala (2014, preprint)