The paper deals with vector-valued inequalities in a setting of weighted Lebesgue measures \(\mu_k\) invariant under the action of a reflection group \(G\). The authors first present a theorem for Banach-valued singular integral operators. More precisely, let \(\mathfrak B_1, \mathfrak B_2\) be two Banach spaces and let \(T: L^r_k(\mathbb R^N, \mathfrak B_1) \to L^r_k(\mathbb R^N, \mathfrak B_2)\) be a bounded operator for some \(12|y-y_0|}\|\mathcal K(x,y)-\mathcal K(x,y_0)\|d\mu_k(x)\leqslant C, \quad y,y_0 \in \mathbb R^N, \] \[ \int_{\mathrm{min }_{g \in G}|g.x-y|>2|x-x_0|}\|\mathcal K(x,y)-\mathcal K(x_0,y)\|d\mu_k(y)\leqslant C, \quad x,x_0 \in \mathbb R^N, \] with \(\mathopen\|\cdot\mathclose\|\) the usual norm of \(\mathcal L(\mathfrak B_1,\mathfrak B_2)\), then \(T\) can be extended to a bounded operator from \(L^p_k(\mathbb R^N, \mathfrak B_1)\) to \(L^p_k(\mathbb R^N, \mathfrak B_2)\) for all \(1