The whole point of widget toolkits is that the keybindings for input fields are the same everywhere. I use Linux and almost exclusively run GTK-based apps, and every text input across the whole system behaves the exact same way, with consistent keybindings for navigation, undo/redo, copy+paste etc. This is how good UI should work.
Platform conventions are important, and it's the job of individual programs to adapt to them – not the other way round. Vim fails big time in this regard. Can you make Firefox's form inputs use Vim keybindings? If not, there's your inconsistency already.