Based on $14.7~textrm{fb}^{-1}$ of $e^+e^-$ annihilation data collected with the BESIII detector at the BEPCII collider at 17 different center-of-mass energies between $3.7730~textrm{GeV}$ and $4.5995~textrm{GeV}$, Born cross sections of the two processes $e^+e^- to pbar{p}eta$ and $e^+e^- to pbar{p}omega$ are measured for the first time. No indication of resonant production through a vector state $V$ is observed, and upper limits on the Born cross sections of $e^+e^- to V to pbar{p}eta$ and $e^+e^- to V to pbar{p}omega$ at the $90%$ confidence level are calculated for a large parameter space in resonance masses and widths. For the current world average parameters of the $psi(4230)$ of $m=4.2187~textrm{GeV}/c^{2}$ and $Gamma=44~textrm{MeV}$, we find upper limits on resonant production of the $pbar{p}eta$ and $pbar{p}omega$ final states of $7.5~textrm{pb}$ and $10.4~textrm{pb}$ at the $90%$ CL, respectively.
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