If you’d like your video to autoplay or loop, go to the video you'd like and copy the URL of the page (not the Share button embed code), as you can read in the oEmbed dedicated documentation. Then just add the default parameters: loop=1 for the loop action and autoplay=1 for the autoplay option. Please note that the video can autoplay only if embedded in the page (not in lightbox):
YouTube: https://youtube.com/video-url?autoplay=1&loop=1 Vimeo: https://vimeo.com/video-url?autoplay=1&loop=1
You can also apply the autoplay and loop option to your self-hosted video as well. In your Media Library just set the Autoplay and Loop option as you can see in this screenshot.
Note on video background
Please note that videos used as backgrounds are designed to have automatic loops. If you use a video as background, it will loop automatically (even without the 'loop' parameter inserted).