In our district, we have deployed close to a thousand devices for which there is no ITAM client agent. We have Chromebooks, Chromeboxes, iPads and Android tablets. I'd like to see the addition of new client agents for operating systems other than Windows and Mac OS.
I think for Chrome OS, it would be simplest to create it as an app or an extension that can be made mandatory through the enterprise management console. It could determine what SchoolDude account to tie it to by looking at the domain of the GAFE enterprise account that manages the device. A similar solution may also work well for IOS and Android operating systems.