Looks like Firefox does not support it currently.
[1] https://github.com/mozilla/standards-positions/issues/154
In place of this, Firefox recommends using the origin private file system (OPFS)[2] which is basically a sandboxed filesystem that the users cannot access. Though I agree from a security perspective it kind of makes sense (not particularly for my usecase), but it removes the ability for the user to have realtime changes on local files.
An example of this is using the web based tool as an intermediary step or a processing step in a bigger pipeline; I save it using my web tool and then native applications automatically handle file updates. This just creates an extra step for Firefox/Safari to replicate changes locally by having to manually update local files.
Also neither Firefox nor Safari support showDirectoryPicker() which also creates blocks for building a web based editor.
I love Firefox and I hope they can add support for this. I have stopped caring about Safari/Webkit completely because it's hellish to try and maintain feature parity while NOT sacrificing innovation and ease of use.
[1] https://developer.mozilla.org/en-US/docs/Web/API/FileSystemO...
[2] https://developer.mozilla.org/en-US/docs/Web/API/File_System...