The production of a Vector Boson in association with a single charm quark is measured and presented here using 140 fb$^{-1}$ of proton--proton collision data collected with the ATLAS detector at the Large Hadron Collider.Measurements are performed using either a Z boson or W boson produced with a charm meson. The charm quark is tagged by the presence of a charmed hadron, reconstructed with a secondary-vertex fit. The $W$ boson is reconstructed from the decay to either an electron or a muon and the missing transverse momentum present in the event. The $Z$ boson is reconstructed from the decay to either a pair of electrons or muons. The reconstruction modes used for charm hadron reconstruction are $D^{+} \to K^- \pi^+ \pi^+$ and $D^{*+} \to D^{0} \pi^+ \to (K^- \pi^+) \pi^+$ and the charge conjugate decays. The \WplusDmeson integrated and normalized differential cross-sections as a function of the pseudorapidity of the lepton from the $W$ boson decay, and of the transverse momentum of the charmed hadron, are extracted from the data using a profile likelihood fit. The fiducial region is defined to include events passing $p_{\text{T}}(e, \mu) > \SI{30}{\GeV}$, $|\eta(e, \mu)| < 2.5$, $p_{\text{T}}(D^{(\ast)}) > \SI{8}{\GeV}$, and $|\eta(D^{(\ast)})| < 2.2$. The measured total fiducial cross-sections are: $\sigma^{\mathrm{OS-SS}}_{\mathrm{fid}}(W^{-}{+}D^{*+}) = 51.1\pm0.4\,\mathrm{(stat.)}\,^{+1.9}_{-1.8}\,\mathrm{(syst.)}\,\mathrm{pb}$, and $\sigma^{\mathrm{OS-SS}}_{\mathrm{fid}}(W^{+}{+}D^{*-}) = 50.0\pm0.4\,\mathrm{(stat.)}\,^{+1.9}_{-1.8}\,\mathrm{(syst.)}\,\mathrm{pb}$. Results are compared with the predictions of next-to-leading-order quantum chromodynamics calculations performed using state-of-the-art parton distribution functions. Additionally, the ratio of charm to anti-charm production cross-sections is studied to probe the $s$-$\bar{s}$ quark asymmetry. The ratio is found to be $R_c^\pm = 0.971\pm0.006\,\mathrm{(stat.)}\pm0.011\,\mathrm{(syst.)}$. The ratio and cross-section measurements are consistent with the predictions obtained with parton distribution function sets that have a symmetric $s$-$\bar{s}$ sea, indicating that any $s$-$\bar{s}$ asymmetry in the Bjorken-$x$ region relevant for this measurement is small. The \ZplusDmeson integrated and normalized differential cross-sections as a function of the transverse momentum of the charmed hadron, are extracted from the data using the same fit strategy. The fiducial region is defined to include events passing $p_{\text{T}}(e, \mu) > \SI{27}{\GeV}$, $|\eta(e, \mu)| < 2.5$, $p_{\text{T}}(D^{(\ast)}) > \SI{8}{\GeV}$, and $|\eta(D^{(\ast)})| < 2.2$. The measured total fiducial cross-sections in the \ZplusDmeson analysis are $\sigma_{\mathrm{fid}}(Z^{0}{+}D^{\pm}) = 21.9 \pm0.04 \,\mathrm{(stat.)}\,^{+1.5}_{-1.4}\,\mathrm{(syst.)}\,\mathrm{pb}$, and $\sigma_{\mathrm{fid}}(Z^{0}{+}D^{*\pm}) =31.1 \pm0.23 \,\mathrm{(stat.)}\,^{+1.3}_{-1.3}\,\mathrm{(syst.)}\,\mathrm{pb}$. These results are also compared with the predictions of next-to-leading-order quantum chromodynamics calculations performed using state-of-the-art parton distribution functions.