  The process $e^+e^- \to \gamma\chi_{cJ}$ ($J$=1, 2) is studied via initial
state radiation using 980 fb$^{-1}$ of data at and around the $\Upsilon(nS)$
($n$=1, 2, 3, 4, 5) resonances collected with the Belle detector at the KEKB
asymmetric-energy $e^+e^-$ collider. No significant signal is observed except
from $\psi(2S)$ decays. Upper limits on the cross sections between
$\sqrt{s}=3.80$ and $5.56~{\rm GeV}$ are determined at the 90% credibility
level, which range from few pb to a few tens of pb. We also set upper limits on
the decay rate of the vector charmonium [$\psi(4040$), $\psi(4160)$, and
$\psi(4415)$] and charmoniumlike [$Y(4260)$, $Y(4360)$, and $Y(4660)$] states
to $\gamma\chi_{cJ}$.
