I want to be able to connect to a linux server with a gui using something similar to x forwarding or no machine's nx client, except have the option for the client to be any browser with no plugins required (i.e. from an ipad/internet cafe pc/friends pc). Is there any software in existence that supports this?
You seem to be looking for NX web player.
VNC can serve up a Java-based client. Most browsers have Java enabled. Just make sure you secure access very well.
The x11vnc program can do what you want. However, I have found that the web-viewer, while functional, was almost unusably slow. Also, it is a desktop-sharing or desktop-display system rather than a remote-access system; whatever you can see on the remote access system is visible on the actual console. This means anyone physically at the computer can see, or potentially can access, anything you can.
X2go has a web accessible interface, but alas it does require a plugin of some form.