  Based on data samples collected with the BESIII detector at the BEPCII
collider at 9 center-of-mass energies from 4.21 to 4.42 GeV, we search for the
production of $e^+e^-\to \omega\chi_{cJ}$ ($J$=0, 1, 2). The process $e^+e^-\to
\omega\chi_{c0}$ is observed for the first time, and the Born cross sections at
$\sqrt{s}$=4.23 and 4.26 GeV are measured to be $(55.4\pm 6.0\pm 5.9)$ and
$(23.7\pm 5.3\pm 3.5)$ pb, respectively, where the first uncertainties are
statistical and the second are systematic. The $\omega\chi_{c0}$ signals at the
other 7 energies and $e^+e^-\to \omega\chi_{c1}$ and $\omega\chi_{c2}$ signals
are not significant, and the upper limits on the cross sections are determined.
By examining the $\omega\chi_{c0}$ cross section as a function of
center-of-mass energy, we find that it is inconsistent with the line shape of
the $Y(4260)$ observed in $e^+ e^-\to\pi^+\pi^-J/\psi$. Assuming the
$\omega\chi_{c0}$ signals come from a single resonance, we extract mass and
width of the resonance to be $(4230\pm8\pm6)$ MeV/$c^2$ and $(38\pm12\pm2)$
MeV, respectively, and the statistical significance is more than $9\sigma$.
