This paper presents multiplicity measurements of $K^0_{\textrm{S}}$, $\Lambda$, and $\bar{\Lambda}$ produced in 120 GeV/$c$ proton-carbon interactions. The measurements were made using data collected at the NA61/SHINE experiment during two different periods. Decays of these neutral hadrons impact the measured $\pi^+$, $\pi^-$, $p$ and $\bar{p}$ multiplicities in the 120 GeV/$c$ proton-carbon reaction, which are crucial inputs for long-baseline neutrino experiment predictions of neutrino beam flux. The double-differential multiplicities presented here will be used to more precisely measure charged-hadron multiplicities in this reaction, and to re-weight neutral hadron production in neutrino beam Monte Carlo simulations.