If you need to know the service tag of a system whilst not physically in front of the machine (or the service tag label is missing!) you can do so via the command line:
If you can RDP to the system, open a command prompt (admin not required) and then enter:
wmic csproduct get vendor,name,identifyingnumber
If you can’t RDP, but are on the same network, you can try:
wmic /user:<USERNAME> /node:<REMOTESYSTEM> bios get serialnumber
Obviously replace <USERNAME> and <REMOTESYSTEM> with the relevant information.
Thanks to Jaysun Burgess for the commands.