Well actually ...
1. The date thing is easy and can be done on a command-line - that part does not require a script.
2. The floating target is more of an issue, it would be best to pin down the target directory so it does not change.
You might give them sudo permissions to run a script (and nothing else) but that is only secure if the script cannot change. Placing it on a read-only partition (cd perhaps) might be wise. Restrict the file name to something hard-coded in, but let it request the folder location from the user.
Make sure that your script evaluates the input to ensure that the folder exists, it does not overwrite an existing file, and that the user is not making a typo that could get someone fired.