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 $s=13$ TeV and recorded by the ATLAS detector corresponds to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. 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 $(W/Z)Z(\to cc¯)$ process and $3.8$ ($4.6$) standard deviations for the $(W/Z)W(\to cq)$ process. The $(W/Z)H(\to cc¯)$ 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 125 GeV, corresponding to an observed (expected) constraint on the charm Yukawa coupling modifier , at the 95% confidence level. A combination with the ATLAS $(W/Z)H,H\to bb¯$ analysis is performed, allowing the ratio $\kappa c/\kappa b$ to be constrained without assumptions about the width of the Higgs boson. The ratio is constrained to be 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.