A search for the Higgs boson decaying into a pair of charm quarks is presented. The analysis uses proton–proton collisions to target the production of a Higgs boson in association with a leptonically decaying W or Z boson. The dataset delivered by the LHC at a centre-of-mass energy of [InlineEquation not available: see fulltext.] and recorded by the ATLAS detector corresponds to an integrated luminosity of 139 fb-1. Flavour-tagging algorithms are used to identify jets originating from the hadronisation of charm quarks. The analysis method is validated with the simultaneous measurement of WW, WZ and ZZ production, with observed (expected) significances of 2.6 (2.2) standard deviations above the background-only prediction for the (W/ Z) Z(→ cc¯) process and 3.8 (4.6) standard deviations for the (W/ Z) W(→ cq) process. The (W/ Z) H(→ cc¯) search yields an observed (expected) upper limit of 26 (31) times the predicted Standard Model cross-section times branching fraction for a Higgs boson with a mass of [InlineEquation not available: see fulltext.], corresponding to an observed (expected) constraint on the charm Yukawa coupling modifier |κc|<8.5(12.4), at the 95% confidence level. A combination with the ATLAS (W/ Z) H, H→ bb¯ analysis is performed, allowing the ratio κ_c/ κ_b to be constrained to less than 4.5 at the 95% confidence level, smaller than the ratio of the b- and c-quark masses, and therefore determines the Higgs-charm coupling to be weaker than the Higgs-bottom coupling at the 95% confidence level.