Android System WebView's update is now released for public. Still, if you want to get (beta) update earlier, then continue reading:
As the article says,
If you're making use of the WebView in your app, becoming a beta channel tester will give you an early start with new APIs as well as the chance to test your app before the WebView rolls out to your users.
The first version offered in the beta channel will be based on Chrome 40 and you can find a full list of changes on the chromium blog entry.
To become a beta tester, join the community which will enable you to sign up for the Beta program; you'll then be able to install the beta version of the WebView via the Play Store. If you find any bugs, please file them on the Chromium issue tracker.
As of current, the WebView is based on Chrome 37, while the beta channel offers Chrome 40. Also, currently there is no update publicly available, hence the Play Store link displaying "Not Found". To be able to see page (and update the WebView), you have to join beta channel testing. The guide as provided on the Google+ community:
- Join WebView Beta Channel community on Google+.
- Sign up as a tester via this link.
- Go to the Play Store page and ensure your devices are updated with the latest version.