Hi Sam-
Video backgrounds are currently only natively supported in our Crio theme, which is also available for free on the WordPress Repository.
It is possible to use a GIF in our other themes, there’s just two things you need to know to make sure it works correctly. While you’re setting up the GIF as your header background, make sure to click Skip Cropping while you’re adding the background image, and use the Scaled to Fit background size option.
One thing I should mention here is that GIF files tend to be very large, and could have a negative impact on your site speed and SEO, so if you go with a GIF you should make sure it’s a well-optimized GIF. A benefit of using the Crio theme for this is that you can use a YouTube video, which won’t block the rendering of your site while it loads.