Yes, this can be done but is of course ill-advised.
First, check to see whether this application has a hard-coded
name for the temp file. If so, try creating a file with that name
and then granting Users group Modify to that one file.
If that does not work, and your C: drive is at its defaults as far
as NTFS permissions go, try this next. Access the security
dialog in the properties of C: itself. Click on Advanced so that
you see the distinct grants. At default values there will be one
to Users granting Special. Highlight this one and click edit. In
the list of checkboxes you will see it grants Create Files/Write Data
and that it is effective for (Applies To) Subfolders only. In the
Applies Onto dropbox change the selection from Subfolders only
to This Folder and Subfolders.
Now, if the application cleans up after itself that should be all
that is needed as the grant to Creator Owner will allow the other
needed permissions. However, if it does not, and it tries to then
reuse a temp file when it is run under a different account, this
will not work. You would then need to grant Users group Modify
on the file.
