If you keep the device awake and the screen active during the download, once downloaded yes you can install the downloaded update without having it plugged in. You cannot complete the download of an iOS update if unplugged the screen asleep as the wifi radio goes off when the screen blanks and the devixe sleeps.
Once downloaded, running the installer without being plugged in will be fine, as long as your battery charge is sufficient to complete the process. If your battery dies mid-installation, you may have to restore the device and then use your backup to get back up and running.
This thread was initially about downloading over wifi without being plugged in, and that has never been possible as when running on battery, the wifi radio switches off when the device enters sleep mode. But there is a distinction between the downloading of an iOS update, and then running that download to install the update. The former requires wifi, the latter does not.