The Web Speech API's Speech Recognition functionality is implemented in some major browers (major Chromium-based ones like Chrome and Edge, as well as Webkit-based ones like Safari).
As of June 2023 / Firefox 114, the MDN page lists "unsupported" for Firefox, and indeed trying to construct a SpeechRecognition object doesn't work, even with the special media.webspeech.recognition.enable flag listed in this Mozilla blog entry from 2016.
However, Firefox internal documentation seems to suggest the feature is in some way present. Looking through Firefox bugs and discussions, bug 1248897 seems to have been the major hub for the implementation of this feature from 2017 to 2021, with over 100 comments, many commits, and eventually a "Status: RESOLVED FIXED".
It seems like it was never fully implemented. However, as of January 2025 Firefox has officially indicated they are "Positive" about implementing the "Web Speech API".
We believe this is a strong and welcome improvement to the API, on multiple aspects: added capabilities to the platform, in terms for feature and performance (e.g. being able to do high-quality low-latency recognition), but also accessibility and privacy.
The GitHub Issue that triggered that change: GitHub: On-device Web Speech API
This is positive and we're actively investigating, initially focusing on the recognition side of things.
I cannot find a timeline of when this will come to Firefox—it is long overdue. However, a "positive" position usually means Firefox plans to implement the API in Firefox.
In fact, the entry in "Mozilla's Standards Positions" has a brand new bug ticket connected to it: Bugzilla: Implement on-device Web Speech Recognition. That bug is marked with P2 priority (ranges from P1 to P5), which means:
"We want this, but it's not totally clear or extremely important."
The full list of Mozilla's positions on standards and proposed standards can be found here: https://mozilla.github.io/standards-positions/#web-speech-api-on-device