The docs on how to use existing javascript web systems within a rust -> wasm flow using wasm-bindgen are pretty straight-forward.  In short:
- List the needed features as dependencies
 
// cargo.toml
[dependencies.web-sys]
version = "0.3.4"
features = [
  'AudioContext',
  'OscillatorNode'
]
- Access through web_sys, and treat them as normal from there.
 
// lib.rs
#[wasm_bindgen]
impl FmOsc {
    #[wasm_bindgen(constructor)]
    pub fn new() -> Result<FmOsc, JsValue> {
        let ctx = web_sys::AudioContext::new()?;
        let primary = ctx.create_oscillator()?;
        let fm_osc = ctx.create_oscillator()?;
Link to full example
While this approach works for most web systems, it is not so simple for ones which require you to extend a class.  In this case I'm trying to use the AudioWorkletProccessor from wasm-bindgen
In standard usage, you are supposed to extend the AudioWorkletProcessor class, then register it. MDN Example
How can I approach this problem?  First I need to extend AudioWorkletProcessor in Rust (which doesn't do normal classes and inheritance), overriding its process() function.  Then I need to make sure registerProcessor() is injected into my rust space.
EDIT: WASM docs on extends
                        
You will have to do that on the JavaScript side.
wasm-bindgendoes not support extending imported classes and re-exporting them to JavaScript; however, they do want to support it eventually and there is an issue to track it, and a related RFC.