etsoft
I do not want to set everything to 777.
.. and you shouldn't; nothing should ever be set to 0777. File permissions are typically 0644 and Folder permissions are typically 0755.
The data doesn't save to a file, it saves to the database. The system simply sends a POST request to a file which calls subsequent code to save the data in the database.
You should try what I mentioned above:
You can also inspect the request to see if the response contains a more specific message other than "Forbidden".
Cheers.