A search is presented for new particles produced in proton-proton collisions at a centre-of-mass energy of 13 TeV that result in final states comprising a massive vector (W or Z) boson that decays hadronically and large missing transverse momentum. The data sample was collected with the ATLAS experiment at the Large Hadron Collider from 2015 to 2018 and corresponds to an integrated luminosity of 140 fb$^{−1}$. No significant excess over the Standard Model expectation is observed. Model-independent 95% confidence-level limits on the visible cross-section that range from 0.3 fb to 79.5 fb are obtained for non-Standard-Model processes. Exclusion limits are also presented for models with axion-like particles, for two-Higgs-doublet models with a pseudo-scalar mediator between the Standard Model and the dark sector, for the invisible decay of the Higgs boson and for pair-produced weakly interacting dark matter candidates.[graphic not available: see fulltext]