Let $K_\lambda^d$ be the convex hull of the intersection of the homogeneous Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, $d \ge 2$, with the Euclidean unit ball $\mathbb{B}^d$. In this paper, we study the asymptotic behavior as $d\to\infty$ of the support function $h_\lambda^{(d)}(u) :=\sup_{x\in K_\lambda^d}\langle u,x\rangle$ in an arbitrary direction $u \in {\mathbb S}^{d-1}$ of the Poisson polytope $K_\lambda^d$. We identify three different regimes (subcritical, critical, and supercritical) in terms of the intensity $\lambda:=\lambda(d)$ and derive in each regime the precise distributional convergence of $h_\lambda^{(d)}$ after suitable scaling. We especially treat this question when the support function is considered over multiple directions at once. We finally deduce partial counterparts for the radius-vector function of the polytope.