This apparently possible, according to this StackOverflow post.
Before posting the content of the answer, however, can I suggest that you're over-complicating this?
In situations like this where some crappy piece of code needs a user logged on to run (like Domino server, grumble) I've created a service account that's to always be logged in on a given server, and setup an auto-login script, so that the machine logs in the specified account on reboot automatically. I'd suggest that the easier, and more supportable solution to your problem would be to do the same, and have the drive mapped for that service account user by Group Policy or logon script.
Anyway, should you be determined to try this without a user context, see the below. It's a hack, so use at your own risk and all:
For this hack you will need SysinternalsSuite by Mark Russinovich: http://technet.microsoft.com/en-us/sysinternals/bb842062.aspx
Step one: Open an elevated cmd.exe prompt (Run as administrator)
Step two: Elevate again to root using PSExec.exe: Navigate to the folder containing SysinternalsSuite and execute the following command "psexec -i -s cmd.exe" you are now inside of a prompt that is "nt authority\system" and you can prove this by typing "whoami". The -i is needed because drive mappings need to interact with the user
Step Three: Create the persistent mapped drive as the SYSTEM account with the following command "net use z: \servername\sharedfolder /persistent:yes"
It's that easy!
WARNING: You can only remove this mapping the same way you created it, from the SYSTEM account. If you need to remove it, follow steps 1 and 2 but change the command on step 3 to: "net use z: /delete"
NOTE: The newly created mapped drive will now appear for ALL users of this system but they will see it displayed as "Disconnected Network Drive (Z:)". Do not let the name fool you. It may claim to be disconnected but it will work for everyone. That's how you can tell this hack is not supported by M$.
To get it working after a reboot, create a script just containing net use z: \servername\sharedfolder and set it to run on computer startup, per technet.microsoft.com/en-us/library/cc770556.aspx This will run as the SYSTEM account, so no need for psexec.