We study theoretically a triangular duster of three magnetic impurities, hybridizing locally with conduction electrons of a metallic host. Such a cluster is the simplest to exhibit frustration, an important generic feature of many complex molecular systems in which different interactions compete. Here, low-energy doublet states of the trimer are favored by effective exchange interactions produced by strong electronic repulsion in localized impurity orbitals. Parity symmetry protects a level crossing of such states on tuning microscopic parameters, while an avoided crossing arises in the general distorted case. Upon coupling to a metallic host, the behavior is shown to be immensely rich because collective quantum many-body effects now also compete. In particular, impurity degrees of freedom are totally screened at low temperatures in a Kondo-screened Fermi liquid phase, while degenerate ground states persist in a local moment phase. Local frustration drives the quantum phase transition between the two, which may be first order or of Kosterlitz-Thouless type, depending on symmetries. Unusual mechanisms for local moment formation and Kondo screening are found due to the orbital structure of the impurity trimer. Our results are of relevance for triple quantum dot devices. The problem is studied by a combination of analytical arguments and the numerical renormalization group.