TL;DR: Yes, it should be possible to access the Leap Motion data in WebGL through the WebSocket interface, although our core assets don't support it.
Now Unity recently added support for WebSockets, though that support is still a bit raw. This could be used to get the Leap Motion data from the WebSocket connection. See:
I tried this approach and was able to get the data. I also refactored the C# interface code to feed the Core Asset scripts with data coming from the WebSocket rather then the LeapCSharp shared library. This actually worked reasonably well in the Editor, but, and it is a big but, exporting to WebGL (or the Web Player) didn't produce anything that ran. I think the reason is that the Core Assets are too complex for the emscripten translation -- or it could be the assets are just too large -- or something else, I didn't have time to investigate further.
Unity has a WebSocket plugin in the Asset Store that may help (I haven't tried it):