  We present measurements of the Born cross sections for the processes
$e^+e^-\rightarrow\omega\chi_{c1}$ and $\omega\chi_{c2}$ at center-of-mass
energies $\sqrt{s}$ from 4.308 to 4.951 GeV. The measurements are performed
with data samples corresponding to an integrated luminosity of 11.0
$\rm{fb}^{-1}$ collected with the BESIII detector operating at the BEPCII
storage ring. Assuming the $e^+e^-\rightarrow\omega\chi_{c2}$ signals come from
a single resonance, the mass and width are determined to be
$M=(4413.6\pm9.0\pm0.8)$ MeV/$c^2$ and $\Gamma=(110.5\pm15.0\pm2.9)$ MeV,
respectively, which is consistent with the parameters of the well-established
resonance $\psi(4415)$. In addition, we also use one single resonance to
describe the $e^+e^-\rightarrow\omega\chi_{c1}$ lineshape, and determine the
mass and width to be $M=(4544.2\pm18.7\pm1.7)$ MeV/$c^2$ and
$\Gamma=(116.1\pm33.5\pm1.7)$ MeV, respectively. The structure of this
lineshape, observed for the first time, requires further understanding.
