Is there an estimated date for when the linux webplayer might be distributed?
No, no date, not even a year or target version.
Its at thing in research / work with undefined timeline at least publically
Unity 4.0 will ship with a preview of a Linux Export feature that will allow you to publish content for 32-bit and 64-bit desktop systems. Please see Unity Pro
We still have no plans to ship a webplayer at this time.
Unity Technologies do not provide a Unity Web Player for Linux browsers. I also haven’t heard any plans on doing so.
Workaround:
Previously Linux users tried to install Windows versions of internet browsers to run the Unity Web Player using Wine.
The Pipelight project has made a plugin for native Linux browsers. It still uses Wine, but at least you won’t deal with Windows versions of internet browsers. See the community set up wiki page - http://wiki.unity3d.com/index.php/Running_Unity_Web_Player_on_Linux_using_Pipelight