We report a measurement of the \(e^+e^- \to \pi^+\pi^-\pi^0\) cross section in the energy range from 0.62 to 3.50 GeV using an initial-state radiation technique. We use an \(e^+e^-\) data sample corresponding to 191 \(\text{fb}^{-1}\) of integrated luminosity, collected at a center-of-mass energy at or near the \(\Upsilon{(4S)}\) resonance with the Belle II detector at the SuperKEKB collider. Signal yields are extracted by fitting the two-photon mass distribution in \(e^+e^- \to \pi^+\pi^-\pi^0\gamma\) events, which involve a \(\pi^0 \to \gamma\gamma\) decay and an energetic photon radiated from the initial state. Signal efficiency corrections with an accuracy of 1.6% are obtained from several control data samples. The uncertainty on the cross section at the \(\omega\) and \(\phi\) resonances is dominated by the systematic uncertainty of 2.2%. The resulting cross sections in the 0.62-1.80 GeV energy range yield \( a_\mu^{3\pi} = 48.91 \pm 0.23~(\mathrm{stat}) \pm 1.07~(\mathrm{syst}) \times 10^{-10} \) for the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. This result differs by \(2.5\) standard deviations from the most precise current determination.