The yields of $K^0_S$ mesons have been measured in inelastic p+p interactions at incident projectile momenta of 31, 40 and 80 $\text {Ge}\hspace{-1.00006pt}\text {V}\!/\!c$ ($\sqrt{s_{NN}}=7.7, 8.8$ and 12.3 $\text {Ge}\hspace{-1.00006pt}\text {V}$, respectively). The data were recorded by the NA61$/$SHINE spectrometer at the CERN Super Proton Synchrotron and the $K^0_S$ mesons identified via their decays into $\pi ^{+} \pi ^{-}$ pairs. Double-differential distributions are presented as function of transverse momentum and rapidity. The mean multiplicities of $K^0_S$ mesons were determined to be $(5.95 \pm 0.19 (stat) \pm 0.30 (sys)) \times 10^{-2}$ at 31 $\text {Ge}\hspace{-1.00006pt}\text {V}\!/\!c$, $(7.61 \pm 0.13 (stat) \pm 0.43 (sys)) \times 10^{-2}$ at 40 $\text {Ge}\hspace{-1.00006pt}\text {V}\!/\!c$ and $(11.58 \pm 0.12 (stat) \pm 0.55 (sys)) \times 10^{-2}$ at 80 $\text {Ge}\hspace{-1.00006pt}\text {V}\!/\!c$. The results on $K^{0}_{S}$ production are compared with the production of charged kaons in corresponding reactions and with model calculations (Epos1.99, SMASH 2.0 and PHSD) as well as with published data from other experiments.