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.
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