Exchange 2013 Mail Relay from External Web Service

All of our mail relaying for notifications are done internally.  Our web host is wanting to relay mail for job application submissions to our HR Department.  They are using to test the SMTP settings I provide them, but fails?  I have created a mail relay user in exchange with a password but still no luck.  I can send an email from an external email client via telnet tho?

Normally from withing the Exchange 2013 Admin Center I will just add the internal IP addresses to a receive connector I have configured to allow internal servers to relay mail for alerts.  Would this be the same for an outside vendor/service?

Is there anywhere I can see logs as to why this is failing or where to double check the configurations?
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AmitIT ArchitectCommented:
Ask your vendor to relay via internet and use its own SMTP server to relay mails.

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