  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 $e^+e^- \to p\bar{p}\eta$ and $e^+e^- \to
p\bar{p}\omega$ 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 $e^+e^- \to V \to p\bar{p}\eta$ and $e^+e^- \to V \to
p\bar{p}\omega$ 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
$p\bar{p}\eta$ and $p\bar{p}\omega$ final states of $7.5~\textrm{pb}$ and
$10.4~\textrm{pb}$ at the $90\%$ CL, respectively.
