Does that mean I can install cargo-web without emscripten or do I still need it even though I am not going to use it just because the two other targets use it? In other words, is emscripten an optional or a hard dependency?

Its an optional dependency. You can build without it by just using a nightly version of rust with the wasm32-unknown-unknown backend.

An example of using cargo-web with the stdweb rust project: https://github.com/koute/stdweb#running-the-examples