Might be me
Sounds like me
So it should work.
There are two ways. One involves using the HTML5 BACKGROUND VIDEO Webflow widget, the other using custom code to embed an HTML5 video and eventually editing the code you'll download to point to the local files.
There is a difference between an HTML5 video background and an HTML5 video. The latter shows the entire video at a given size, the former fills an element (DIV) with the video, which—depending on the DIV ratio—can crop the video a little bit. But using the HTML5 Background Video widget is easier as the video will be downloaded with the site and you won't have to edit the HTML code to make sure the HTML page points to the video file.
If having the video autoplay and loop at the loading of the page is OK for you I suggest you try the widget solution first, as it's simpler, and if it fits your needs, well great, and if it doesn't then use the other option.
Also dimension your widget element with the same ratio than your video. If it's an HD video the ratio is 16:9 so for example dimensions of 320x180 or 640x360 are OK.