Windows provides the Roaming Application Data folder and the Local Application Data folder specifically for use by applications. If the system administrator has locked these down, it probably means that the users aren't supposed to be running third-party applications. (The only other obvious possibility being that the system administrator is incompetent.)
Other possibilities include the Documents folder ... basically any of the per-user folders are expected to be writable under normal circumstances. Of course, the system administrator can make any of these unwritable. But that shouldn't be your problem.
Note that you should be using the SHGetKnownFolderPath function or equivalent to determine the folder path, not reading an environment variable. You haven't specified your language, but there are lots of already answered questions about how to do this.
Also, you shouldn't write files directly to the application data folders; instead, create a subfolder and put the files there.