Normally, if a user can access the SharePoint site through the web, he or she can also login from E-mail Converter. However, that does not apply if the user is invited to log in with a Microsoft account.
Unfortunately Microsoft has not made it possible to authenticate such external users in client side code, so login with a Microsoft account is not possible via E-mail Converter or other client side applications.
We hope that Microsoft will soon do something about this limitation, but for now you have to give SharePoint login credentials to all who should be allowed to log into your SharePoint site via E-mail Converter. Use the SharePoint permissions if you want to restrict access to other parts of the SharePoint site than the E-mail Converter list(s).