I have a laptop which I do not have Administrative privileges for
A lot of networking commands or methods we use require administrative privileges.
Is this a laptop from your work? Why don't you have administrative privileges?
I am wondering if I partitioned an amount of my Desktop Hard Drive I could access files or programs I put on that part of the drive from my laptop.
Depends what programs you're planning on running...
Programs that need installation will probably not work correctly.
What kind of server would I have to set up? Would this work over a VPN? Would I be able to actually run the programs on the partitioned drive from my laptop or just access the raw file?
If you're thinking only about sharing documents then a simple share on your desktop would be enough.