I think a major limitation is that a WebAssembly module can’t run dynamically generated code, which is a huge part of typical Common Lisp implementations.
I was thinking about this, and one solution that came to my mind was serializing the heap into a temporary image and restarting the environment from an updated Wasm module with new code. This would also collect all garbage at the point of transitioning to the new code. You might need to be very careful to do this reasonably quickly, but in principle I don't see why this wouldn't work. The idea came to me when I was thinking about how an interactive-but-native-speed Oberon environment could be implemented in Wasm.
On your computer, heap, stack, code, and global data all share the same address space. On the WASM virtual machine, only heap and global data share the same address space. Code and stack are abstracted away by the machine.
It was designed to support C-like languages, so you can do the equivalent of loading a DLL (or dynamic shared object). However in Lisp it is not unusual to compile just one function at a time. It's definitely possible to create a DLL for each function and load each DLL, but it requires a completely different architecture than most Lisp implementations use (in which, when you compile a function, it just writes the machine code to the heap).
There are similar impedance mismatches with so-called W^X systems (where you are disallowed from having a page be both (W)ritable and e(X)ecutable.