You need the Dell service tag number for a computer, but the machine is not currently in tech’s possession. However, it is up and on the domain.
From a command prompt (does not need to be admin), type:
wmic /user:\administrator /node: bios get serialnumber
Substitute the remote computer’s name for <domain> if logging in as local administrator.
It will prompt for the administrator’s password for the remote computer, and, hopefully, spit out the info you need.
Network issues can interfere with this, though, so if you don’t succeed at first, try again at a later time.