Yes! It does, but at the moment you need to make a few tweaks. These will hopefully change in the future as Mullvad improves.
- Install the Firefox Strongbox AutoFill extension in your Mullvad browser
- Allow Strongbox to "Run in Private Windows" under Strongbox extension settings, from the Manage Extensions screen in Mullvad Browser.
- Enable "webextensions.storage.sync.enabled" in about:config in Mullvad Browser.
We've been in touch with the folks over at Mullvad and hope things will be less clunky in the near future.