Let $P = \{x \in \mathbb{R}_{\geq0}^n \colon Ax \leq b\}$ with $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R^m}$. Let $E \subseteq P$ be the extreme points of $P$.
Can anything be said about the minimum value of $\min_i (|p^1_i - p^2_i|)$ with $p^1,\,p^2 \in E$ without enumerating the whole set $E$? (Solving some auxiliary linear program to compute this value or some lower bound is also fine.)
In general I'm almost certain that there is no way to get the exact answer to your optimization problem without enumerating all vertices (extreme points). The reason why is that there are infinitely many ways to write two systems of linear equations whose solutions have one coordinate equal, and the angles between the hyperplanes for the equations can essentially be whatever you want. So how can you predict whether/which pair of systems of equations will give an identical coordinate, unless you search all solutions to valid systems of equations? (I.e. enumerate all extremal points.)