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Scalar fields coupled to dark matter by conformal or disformal transformations give rise to a general class of scalar-tensor theories which leads to a rich phenomenology in a cosmological setting. While this possibility has been studied comprehensively in the literature for scalar fields, the vector case has been hardly treated. We build hence models based on vector fields conformally and disformally coupled to dark matter and derive explicitly the general covariant form of the interaction term in an independent way of the gravity theory, whereby this result can be applied to general vector-tensor theories. For concreteness, the standard Proca theory with a vector exponential potential is taken to describe the vector-tensor sector, and some specific coupling functions are assumed to study the cosmological background dynamics by dynamical system techniques. Despite of choosing such a minimalist form for the underlying theory, the parameter space is considerably enriched compared to the uncoupled case due to the novel interactions, leading to new branches of solutions for the vector equation of motion. Thus, different trajectories can exist in phase space depending on the coupling parameters associated to the conformal and disformal functions. From here, new emerging vector-dark matter scaling solutions, and renewed stable attractor points are found to drive the late-time accelerated expansion of the universe. As a first examination about instabilities issues, we derive general conditions to avoid classical instabilities in a more general setup of the theory. Numerical calculations are performed as well to investigate more quantitatively the impact of the conformal and disformal couplings on the cosmological background evolution. These effects depend essentially on the strength on the coupling parameters and, in some specific cases, on their associated signs.