You can use a "robot" FM file to do this.
Create a stripped down fmp file.
Write a simple script:
Perform Script [Specify the script in the hosted FMP file that does the export]
Exit Application //closes the robot file
Define an account name and password that matches an account name and password in the hosted file that has sufficient access to run your export script.
Use File Options in the Robot File to:
Set the file to open itself with the above described account and password
Run the the script when the file opens.
Install this file on a secure machine (I install FMP on the server and place this file on the server--since it runs after hours, it doesn't affect performance.) and use the OS to schedule when the file will open. On Windows systems, you can use Task Manager. In Mac systems, I believe there's a widget called "Chronos" or some such that can do the same.
Note: one thing to be careful of is that if there are any problems like an infinite loop in the script being called by the robot file, it will continue to run even after the robot file has closed. I once had to close and reopen the db file on the server to interrupt such a script.