Let \(E\) be a complete, barrelled locally convex space and let \(HV(D,E)\) be the weighted space of holomorphic, vector-valued functions \(D\to E\), where \(V=(v_n)\) is an increasing sequence of strictly positive, radial weights on the open unit disc in the complex plane \(D\). Every analytic map \(\varphi: D\to D\) induces a composition operator \(C_\varphi: f \to f \circ \varphi\), and the purpose of this paper is to characterize when \(C_\varphi\) is reflexive (i.e., it maps each bounded set into a relatively weakly compact set) or weakly compact (i.e., it maps some zero-neighborhood into a relatively weakly compact set). The first step is to prove a result of independent interest, namely that, under mild conditions, \(HV(D,E)\) can be identified with the space \(L(P,E)\) of continuous linear operators from \(P\) into \(E\), where \(P\) is the predual of the corresponding space \(HV(D)\) of scalar-valued functions. This identification yields a representation of \(C_\varphi\) as a wedge operator (i.e., an operator of the form \(T \in L(E_2,E_3) \mapsto L T R \in L(E_1,E_4)\) where \(E_1,\dots, E_4\) are locally convex spaces and \(R\in L(E_1,E_2)\), \(L\in L(E_3,E_4)\) are given operators). The second step is a detailed study of when wedge operators are reflexive or weakly compact. These steps lead to the main result of this interesting paper, stating that if \(C_\varphi : HV(D) \to HW(D)\) is continuous, then (i) \(C_\varphi : HV(D,E) \to HW(D,E)\) is reflexive if and only if \(E\) is reflexive and \(C_\varphi : HV(D) \to HW(D)\) is reflexive; and (ii) \(C_\varphi : HV(D,E) \to HW(D,E)\) is weakly compact if and only if \(E\) is a reflexive Banach space and \(C_\varphi : HV(D) \to HW(D)\) is (weakly) compact.