Did you try adding

UserName ALL= NOPASSWD: /sbin/reboot

As the last line of their /etc/sudoers files?
(replacing UserName with their actual user name, of course.)

That should grant them root access to only the /sbin/reboot command (add
more commands using comma delimiting).

Then they just run
$ sudo reboot

If you want to require they enter their own password before rebooting, use
the line

UserName ALL=/sbin/reboot


Use of sudo is recorded in /var/log/secure, if I recall correctly.

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