Configure SMTP in IIS, so it relays to a remote SMTP server

Here are the steps to set this up using only Microsoft SSRS/Windows Server 2003 components.

1) Install SMTP server in Add/Remove programs / App Server / IIS
In IIS Manager:
2) add the domain (as a remote type) you will be sending as such as yourwebsite.com
3) Under Default SMTP Virtual Server properties / delivery tab / advaced button, add your remote smtp server in the smart host field.
4) Under Default SMTP Virtual Server properties / delivery tab / Outbound security button, choose Basic Authentication, user name = SMTP user name, password = SMTP password

In the rsreportserver.config file (sql drive:\Program Files\Microsoft SQL Server\MSSQL.3\Reporting Services\ReportServer):
5) Populate <SMTPServer>x.x.x.x</SMTPServer> with the IP of the machine where you just setup the SMTP server.
6) Populate <From>you@yourwebserver.com</From>
7) Change this one to false <SendEmailToUserAlias>False</SendEmailToUserAlias>
8) Lastly, make sure you setup the domain as a permitted host such as this:
<PermittedHosts>
<HostName>yourwebsite.com</HostName>
</PermittedHosts>

As far as why SMTP basic authenication isn't supported directly in SSRS seems to be by design. The best explaination I could find was here on this MSDN forum:

As one poster here mentions, there is a Microsoft Connect ticket open for people who are requesting this functionality.

using System.Net;
using System.Net.Mail;

using(SmtpClient smtpClient = new SmtpClient())
{
    var basicCredential = new NetworkCredential("username", "password"); 
    using(MailMessage message = new MailMessage())
    {
        MailAddress fromAddress = new MailAddress("from@yourdomain.com"); 

        smtpClient.Host = "mail.mydomain.com";
        smtpClient.UseDefaultCredentials = false;
        smtpClient.Credentials = basicCredential;

        message.From = fromAddress;
        message.Subject = "your subject";
        // Set IsBodyHtml to true means you can send HTML email.
        message.IsBodyHtml = true;
        message.Body = "<h1>your message body</h1>";
        message.To.Add("to@anydomain.com"); 

        try
        {
            smtpClient.Send(message);
        }
        catch(Exception ex)
        {
            //Error, could not send the message
            Response.Write(ex.Message);
        }
    }
}

You may use the above code.