If you’d like your video to autoplay or loop, go to the video you'd like an copy the URL of the page (not the Share button embed code), as you can read in the oEmbed dedicated documentation. Than 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 embed 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 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.