A search for the Standard Model Higgs boson decaying to a pair of charm quarks is presented. The analysis targets the associated production in proton-proton collisions of the Higgs boson with a $W$ or $Z$ boson decaying into leptons. The dataset delivered by the LHC at a centre-of-mass energy of $\sqrt{s}=13$ TeV 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 hypothesis for the $(W/Z)Z(\to c\bar{c})$ process and 3.8~(4.6) standard deviations for the $(W/Z)W(\to cq)$ process. The $(W/Z)H(\to c \bar{c})$ search yields an observed (expected) limit of $26$ ($31$) times the predicted 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 |\kappa_c| < 8.5~(12.4), at the 95% confidence level.

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