So I accidentally opened an mkv video file with Chrome, and to my surprise it played it using the native player:
I'd be surprised if browsers didn't support mkv, as the webm container is just a restricted version of the matroska format. Beyond that you'd have to check the codec of the video inside, and ensure your browser can handle that. If the video format is VP8 or VP9 and the audio is Opus or Vorbis it's webm compliant and might as well be renamed as such. Note that H.264, H.265 and MP3 are not webm compliant and you should avoid using that extension on files that include those formats, the video would still play but it would technically break the webm specification and is frowned upon.