Well ideally it runs where ever your backup is running so it knows when the backup is done - i assume? Is that on vCenter? how does vCenter notice that the backup is done?
I would assume the easiest option would be to run a "post backup" job (scheduled by the backup tool) to then shut down the box (and set maintenance if you're so inclined).
Whether that script is triggering a shell command on the ESXi box or calls a function at vCenter via api is an implementation detail