Finding out who sent a message on the Facebook Pages App?

Probably not. Given the size of your staff, you should be able to get to the bottom of it or at least eliminate some suspects.

And then limit the number of people who can use the page to communicate with customers. While you might or might not bar the culprit from using the system, you will send a message that you won't tolerate nonsense. Also, create a company policy requiring either documentation or a signature with every communication.

The barn door may be open, and the horse may be gone, but there's no reason to give up and let the cows escape, too.

Thank you @jespah
I appreciate you taking the time to get back to me. Hopefully we'll get to the bottom of it.
The only reason I asked this question is because I'm sure not so long ago the app told you at the bottom who sent the message ie. 'Mark replied 22.18pm' but it no longer does which is strange and annoying as it would help resolve this problem and was also helpful to know which staff are responsible for which customer etc