While cairo-rs is compatible with WebAssembly, it doesn't implement any bindings to it. Instead, create a new WebAssembly crate with cairo-rs as a dependency and implement the required functionality there.
Since mimalloc is not automatically compilable to WebAssembly, the cairo-rs dependency should disable the default features, which will in turn disable mimalloc.
WebAssembly doesn't support filesystem access unless building with WASI support, therefore cairo_run may not work as is. Running programs requires manual program, vm and runner initialization.
A working example is provided in this repository.
Building
# The web target generates a JavaScript module that is directly loadable by the
# browser.
wasm-pack build --target=webRunning
# Running from file:// will result in a CORS error.
wasm-pack build --target=web