It's neither directly related to the web, nor is it an assembly syntax.
It's just another virtual ISA. "Direct DOM access for WASM" makes about as much sense as "direct C++ stdlib access for the x86 instruction set" - none ;)
So you either create a very concrete JS library that translates specific WASM data into IO actions, or one that serializes and deserialize everything all around but can be standardized.
At this point, none of those options are much more capable than Java applets... Or, in fact, if you put a network call between the WASM and the JS, you won't even add much complexity.
Java applets allowed to load and call into native DLLs via JNI, so they were definitely much more capable than WASM, but also irresponsibly unsafe.
In your own WASM host implementation you could even implement a dlopen() and dlsym() to load and call into native DLLs, but any WASM host which cares about safety wouldn't allow that (especially web browsers).
WebASM is an assembly-like dialect, after all.