YouTube, Dailymotion and other websites already allow you to view HTML5 videos, but they don't come with a full-screen button.
For Google Chrome / Chromium, there's an extension which easily fixes this by adding a button where the old full-screen button used to be on YouTube:
(full-screen button added in the bottom right corner)
Here is how the video controls looked before using the Chrome / Chromium HTML5 full-screen (button) extension:
The extension also adds a button in the Omnibar (where applicable) for the rest of the websites which support HTML5 such as Dailymotion and so on:
Once you click the full-screen button, the video will open as full-screen in a tab so if you want to go truly full screen, press F11.
On it's page @ Chrome Extensions, there are also links to install a full-screen button to be placed to the right of the Omnibar as well as context menu (right click) full-screen option.