A search for the standard model Higgs boson decaying to a charm quark-antiquark pair, H $\to \mathrm{c\bar{c}}$, produced in association with a leptonically decaying V (W or Z) boson is presented. The search is performed with proton-proton collisions at $\sqrt{s}=$ 13 TeV collected by the CMS experiment, corresponding to an integrated luminosity of 138 fb$^{-1}$. Novel charm jet identification and analysis methods using machine learning techniques are employed. The analysis is validated by searching for Z $\to \mathrm{c\bar{c}}$ in VZ events, leading to its first observation at a hadron collider with a significance of 5.7 standard deviations. The observed (expected) upper limit on $\sigma(\mathrm{VH}) \mathcal{B}( \mathrm{H \to c\bar{c}} )$ is 0.94 (0.50$^{+0.22}_{-0.15}$) pb at 95% confidence level (CL), corresponding to 14 (7.6$^{+3.4}_{-2.3}$) times the standard model prediction. For the Higgs-charm Yukawa coupling modifier, ${\kappa_{\mathrm{c}}}$, the observed (expected) 95% CL interval is 1.1 $< |{\kappa_{\mathrm{c}}} | <$ 5.5 ($|{\kappa_{\mathrm{c}}} | <$ 3.4), the most stringent constraint to date.

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