> Does someone know if WebAssembly support is available on the open-source Qt?

it is. I know that on windows the toolchain is available directly from the online installer, but I think that it isn't on linux. It's a quick build however :

    git clone qt5 https://code.qt.io/qt/qt5.git
    cd qt5 
    git submodule update --init qtbase qtdeclarative 
    mkdir ../build
    cd ../build
    ../qt5/configure -xplatform wasm-emscripten -nomake examples -nomake tests -opensource -confirm-license -prefix /opt/qt5-wasm
    make -jwhatever && make install