What if you just start with a a simple script like this:
Does that fail?
exec("echo 'Yes, I can run this!'");
I don't use zpanel or anything similar, so I can't say how that might interfere.
Any "../" constructs in the path name of the script being invoked?
Maybe the problem is that the www-data doesn't have privileges to run whatever is being invoked in exec()? I would think you would get a different error message, but it's worth checking like this:
Does it run using this method?
[enter your password]
I searched to see if I could find anything about running shell scripts from Wordpress, but didn't get very far.
Have you checked /var/log/apache2/error.log? Does it help?