Consider a parametric affine variational inequality $0\in Mx + q + N(x; \Delta(A; b))$; denoted by AVI$(M, q, A, b)$; for which the pair $(q, b) \in \mathbb R^n \times \mathbb R^m$ describes the linear perturbations. Here the matrices $M \in \mathbb R^{n\times n}$ and $A \in \mathbb R^{m\times n}$ are the given data, $\Delta(A, b) = \{x \in \mathbb R^n : Ax \leqslant b\}$ is a polyhedral convex constraint set, and $N(x; \Delta(A, b))$ denotes the normal cone to $\Delta(A, b)$ at $x$. We study the normal coderivative of the normal-cone operator $(x, b) \mapsto N(x; \Delta(A; b))$. In the second part of this paper [20], combining the obtained results with some theorems from Mordukhovich [11], Levy and Mordukhovich [10], Yen and Yao [21], we get sufficient conditions for the Aubin property (the Lipschitz-like property) and the local metric regularity in Robinson’s sense of the solution map $(q, b) \mapsto S(q, b)$ of the problem AVI$(M, q, A, b)$ and of the solution map $(w, b) \mapsto S(w, b)$ of the problem $0 \in f(x, w) + N(x; \Delta(A, b))$ where $f : \mathbb R^n \times \mathbb R^s \to \mathbb R^n$ is a given $C^1$ vector function. Our investigation complements the well-known work of Dontchev and Rockafellar [3] where the Aubin property of the solution maps $q \mapsto S(q, b)$ and $w \mapsto S(w, b)$ ($b$ is fixed) was established via a critical face condition.