In order to get SSRS to use Windows Authentication to setup a connection with your local SMTP, you’ll need to modify the rsreportserver.config.
The ‘rsreportserver.config’-file is located in:
X:\Program Files\Microsoft SQL Server\MSRS1111.MSSQLSERVER\Reporting Services\ReportServer
Open the file and look for ‘SMTPAuthenticate‘:
By default it is empty, but you’ll need to set it to ‘2‘.
More info about this property: https://msdn.microsoft.com/en-us/library/ms159155(v=sql.120).aspx#bkmk_options_local_SMTP
SMTPAuthenticate specifies how the report server connects to the remote SMTP server. The default value is 0 (or no authentication). In this case, the connection is made through Anonymous access. Depending on your domain configuration, the report server and the SMTP server may need to be members of the same domain.
To send e-mail to restricted distribution lists (for example, distribution lists that accept incoming messages only from authenticated accounts), set SMTPAuthenticate to 2.
IMPORTANT: After you’ve saved the configuration-file, don’t forget to restart the SSRS service on your server.