The culprit is XDG Desktop Portals. Firefox tries to detect whether it's running inside a sandboxed environment like Flatpak, and when it thinks it is, it hands off file chooser dialogs to the XDG portal service instead of opening a native GTK window. Distrobox leaks enough container-like environment variables to trigger this heuristic, so Firefox dutifully tries to call the portal over D-Bus — but the portal either isn't reachable from inside the container or isn't running at all. The call fails silently and you never see a dialog.
The fix is a single preference change in about:config:
widget.use-xdg-desktop-portal.file-picker = 0
Setting this to 0 tells Firefox to never use the portal for the file picker, falling back to the plain GTK file chooser. The dialog comes back immediately, no restart required.
The default value of 2 means "auto-detect", which is the right call for a real Flatpak sandbox where portals are properly set up. In a Distrobox environment though, the detection fires incorrectly and there's no graceful fallback. Until Distrobox masks the relevant environment variables or Firefox improves its detection logic, flipping this preference manually is the most reliable workaround.