Many remote access solutions provide web access. Most of them use some proprietary plugin. Some work only on local network.
If you want it to be usable on any client, you need a solution that can work over http(s), and does not need any plugin. With html5, you have the chance.
See:
https://github.com/kanaka/noVNC[
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http://freerdp.net/[
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Still, websocket is no drop in replacement for amy socket. Thus you have to consider some server side bridging between websocket and the legacy protocol.