The problems with that are:
1. It means every web app basically has to push its own GC down the wire every time the user hits myawesomeapp.com. That's a lot of overhead.
2. You'd have to define what the memory model between the code running in the VM and the browser's native stuff is. What does it mean for the code to have a reference to a DOM element? How does the browser hold a reference to an event handler in the VM?
3. The set of people who can implement an efficient GC is very very small. Forcing each source language to do that themselves wastes a lot of effort. (For example, look at how primitive Python and Ruby's GCs were for most of their histories.)
Now a low-level VM with GC. That's something I'd be excited about.