Chrome Extentions Autoupdate
Assuming I understand your question correctly, there is no way for software to autoupdate without a device being switched on. Autoupdates require an internet connection to function properly. 

More specifically, an internet connection is needed to compare the currently installed version with the latest version available and, when a new version is available, to download the new version. In some cases, an internet connection is also required to perform the actual installation.

Since an internet connection is required, the device must be powered up for the process to work. In other words a device that is switched off can not autoupdate any of its software, nor can a device that is switched on but not connected to the internet (often referred to as standalone devices).