How to auto invite users to portal when they email in?

[Matt]

As above, when a customer emails into the helpdesk for the first time, we would like an invite to the portal be sent out automatically. 

When a new user emails in I can see the status set to "invited" but the user does not get an invite email.

TIA

[Shikha Verma]

Hi @Matt, Welcome to the community!
Please check Customer Permissions: Anyone can email the service desk or raise a request" is enabled or open. 
Check Notification Settings: Customer-invited notification is enabled
Check if the User is Invited: Go to User management to see if the user status is "Invited". If so, Resend the Invitation
Automation Rules: You can also try automation for the same.