I'm wondering if it's possible to access a LOG or something ?
Maybe - it depends on what the MTA (mail transport agent) is, how it's configured and whether you have the permissions to access the file.
I've got this app running on a GoDaddy server
Then the above are all questions for your GoDaddy provider.
The MTA log should record details of messages which the MTA failed to offload elsewhere but the elsewhere may be a smart relay (i.e. still on the sending side) - in which case you'd need to look at the relay MTA's logs to see if it managed to pass the message on to one of the recipient MXs. Even if the message got as far as a recipient MX, you don't know that it was successfully delivered. By convention, the receiving system should send back a DSN if it fails (and you can specifically ask for a DSN regardless of outcome) but a lot of systems don't implement this properly.
Regardless it may provide information about some messages which fail - you'd need to poll the return mailbox to identify failures.
C.