That's unlikely to happen for a couple of reasons.
Rockbox Utility is based on Qt, so first we need Qt in Chrome OS. As far as I know this doesn't exist. Then we use some C / C++ libraries that would need to run on Chrome OS as well. And last, we need direct hardware access (i.e. root permissions on Linux.) As far as I understand this isn't possible (and not wanted by) Chrome OS.
Especially since Rockbox Utility is mosty a one-shot tool I don't think there will be much interest in working on that, and it's not too much of a problem to get access to a Windows machine these days. But if someone wants to do that ... Rockbox is open source (so we don't need some"open source edition" thing
)