I ran into this issue yesterday - I am currently running Office 365 in a small batch of users for my environment primarily for using SharePoint Online. We do not currently use Azure AD sync or Exchange Online by choice and run our Exchange on-premises. Office 365 users were unable to use Email alerts from SharePoint Online due to their Azure AD records not having email addresses associated to their records since their accounts were created without Exchange Online rights, so email field was not populated. Their usernames were their email addresses, so their alternate emails could not be populated as such.
I tried to use a Microsoft KB article to allow users to edit email information in the SharePoint Online Delve portal, but this KB was dated and did not work: https://docs.microsoft.com/en-us/sharepoint/support/sites/do-not-have-an-e-mail-address-when-create-...
Ultimately I self-resolved the issue by temporarily giving Exchange Online access to the users, waiting 1 hour for replication, then removing Exchange Online. I had to remove Exchange Online due to not being in hybrid configuration (again, by choice) since with Exchange Online activated it would not deliver email to my public MX record and was instead using the self-contained and unused Exchange Online tenancy to deliver email to an unused office 365 mailbox.
All of this could have been avoided if the email address field could be simply be updated in the Azure AD portal, which I could not find a way to do. It would be very beneficial if this field was editable/updateable by a tenant admin to avoid issues like this.
