If it is a browser-based API, there’s no reason it shouldn’t work here. However, many new APIs like this one require your site to have a SSL/TLS certificate (normally there’s an exception made for the localhost domain) [see note below].
Reading the MDN article makes me think that: 1) there is no need for an ssl/tls cert to use this API; 2) maybe it is currently not possible to use the SpeechRecognition API here when using Chrome as your browser. Did you set up any server-side processing in your local development environment? Or just browser APIs and whatnot?
I personally have never used this api (or heard of it for that matter), and it seems like it’s still in an experimental state.