You can use ffmpeg to convert the file on the device if necessary, but integrating ffmpeg is not trivial and converting a video on the device is compute intensive so will take time and use up your battery. If it is possible to change the format server side, it is generally much easier.
If you do want to use ffmpeg, a wrapper approach my be useful - this project provides an example: https://github.com/jhotovy/android-ffmpeg. I have used a similar approach and it works fine.
Make sure in particular you note the comments about calling ffmpeg twice.
There are players in Google play that will play mov files, but not everyone has had good experience with them (for example: https://stackoverflow.com/a/27006587/334402). VLC in particular is a common choice and projects exist to integrate it into Android apps (although maybe not via web views) e.g.: https://github.com/mrmaffen/vlc-android-sdk