Summary: For a fixed nonnegative integer \(m\), an analytic map \(\varphi\) and an analytic function \(\psi \), the generalized integration operator \(I^{(m)}_{\varphi,\psi}\) is defined by \[I^{(m)}_{\varphi,\psi} f(z) = \int_0^z f^{(m)}(\varphi(\zeta)) \psi(\zeta) \, d\zeta\] for \(X\)-valued analytic function \(f\), where \(X\) is a Banach space. Some estimates for the norm of the operator \(I^{(m)}_{\varphi,\psi} \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)\) are obtained. In particular, it is shown that the Volterra operator \(J_b \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)\) is bounded if and only if \(J_b \colon A^2_{\alpha} \to A^2_{\alpha}\) is in the Schatten class \(S_p(A^2_{\alpha})\) for \(2 \leq p < \infty\) and \(\alpha > -1\). Some corresponding results are established for \(X\)-valued Hardy spaces and \(X\)-valued Fock spaces.