Check out the Guacamole project:
Guacamole is a clientless remote desktop gateway. It supports standard protocols like VNC and RDP. We call it clientless because no plugins or client software are required. Thanks to HTML5, once Guacamole is installed on a server, all you need to access your desktops is a web browser.
On the downside, it requires a Java server to proxy through. However, this should be trivial, even if you don't already have a Java server up and running (Tomcat is pretty simple, and Guacamole's instructions are straightforward):
Guacamole is separated into two pieces: guacamole-server, which provides the guacd proxy and related libraries, and guacamole-client, which provides the client to be served by your servlet container, usually Tomcat.
guacamole-client is available in binary form, but guacamole-server must be built from source. Don't be discouraged: building the components of Guacamole from source is not as difficult as it sounds, and the build process is automated. You just need to be sure you have the necessary tools installed ahead of time. With the necessary dependencies in place, building Guacamole only takes a few minutes.