The PDF download URL is always the same. So you could try to download a pdf and attach it to mail via a script (vba, bash, perl...)
You can do this for a PNG image of the dashboard via VizAlerts, A Data-Driven Alerting Tool for Tableau Server. PDF support doesn't exist yet, though, so if that's a hard requirement, it wouldn't work for you (yet, at least).
we've been working on a solution to achieve this and are now in beta.
With our tool you can easily select the contents you want to send out, define the format (CSV, image, PDF, twb, etc.) and schedule the sendout to a distribution list.
Would love to learn your exact requirements.