I believe your SMTP server seems to be blocking the relay for some reason. Your SMTP server administrator needs to analyze the SMTP server logs.
To check if NPrinting itself is sending out the e-mails just fine you can use a dummy SMTP server like SMTP4DEV smtp4dev - Home
HTH - Daniel.
Евгения, Did you resolve your issue?
I've just fixed a similar (?) issue on my side and it was a problem with User's level accounts.
The distribution worked well only for Admin accounts, so additional setting was required in my case.
Logs were misleading, so I am just wondering that you might have the same issue.