AFAICT WebAssembly VM is implemented in native code in the major browsers (Firefox, Chrome, Safari, Edge). It's pretty fast, and has a pretty limited interface to JS or DOM.
Native code or not shouldn't matter that much for a JIT. I've heard Cliff Click say in a talk that if he were to write another VM, it'd probably be in something like Java. The memory management of fairly arbitrary graphs like you get in a compiler really benefit from garbage collection, and it's nice to run the compiler sort of through itself to get dynamic profiling for optimization.
I mean, implementing a WebAssembly VM inside e.g. JVM is both possible and practical, but it's a very different approach from implementing it in native code, let alone any JIT approaches.