We report searches for the processes $e^+e^-\to\pi^+\pi^-\pi^0\chi_{bJ}$ and $e^+e^-\to\phi\chi_{bJ}$ (J=1,2) in the vicinity of the $\Upsilon(11020)$ resonance using energy scan data collected by the Belle experiment at the KEKB collider. We observe $e^+e^-\to \pi^+\pi^-\pi^0\chi_{b1}$ and find evidence for $e^+e^-\to \omega\chi_{b2}$ processes for data with center-of-mass energies from 10.96 to 11.05~GeV for the first time. The statistical significance for $\pi^+\pi^-\pi^0\chi_{b1}$ and $\omega\chi_{b2}$ are $6.1\sigma$ and $4.0\sigma$, respectively. We investigate energy dependence of the $e^+e^-\to\pi^+\pi^-\pi^0\chi_{bJ}$ cross section and find that it is consistent with production of $\Upsilon(10860)$ and $\Upsilon(11020)$ resonances without significant non-resonant contribution. Assuming $e^+e^-\to \pi^+\pi^-\pi^0\chi_{bJ}$ proceeds via $\Upsilon(10860)$ and $\Upsilon(11020)$, the branching fraction $\mathcal{B}(\Upsilon(11020)\to\pi^+\pi^-\pi^0\chi_{bJ})=(8.6\pm4.1\pm6.1^{+4.5}_{-2.5})\times10^{-3}$ is measured for the first time, where the first error is statistical, the second is systematic, and the third from the branching fraction of $\Upsilon(11020)\to e^+e^-$. The signals for $e^+e^-\to \phi\chi_{bJ}$ are not significant, and the upper limits of the Born cross sections at the 90\% confidence level are $0.6$ and $1.0$~pb for $e^+e^-\to \phi\chi_{b1}$ and $\phi\chi_{b2}$, respectively, for center-of-mass energies from 10.96 to 11.05~GeV.