What website template companies (like Wix, Squarespace etc) have a way for me to make a website with a simple moving image in the background, not a still?

What you need to do is use the site building platform you like (choose from this list: ) and after you've done that, integrate a third party gallery widget that does the movement for you. ;) We've provided some sample videos in the link above, but you can see how we integrated one in Slides, for example: The idea is that you'd be using these website building platforms for the structure of your site, but then you can expand the functionality of your sites with widgets as in our example, from https://widgetic.com or from other places.