Mp4 with H.264 encoding is the (more or less) universal standard for web video (.mov is not universally supported on the web).
According to the best information available, H.264 does not support transparency.
Therefore, video with transparency is not directly supported.
However, if your video doesn't have any audio, then rendering it as a PNG sequence and creating an Image Sequence would maintain the transparency.
See this article on Overlays (Widgets) to learn more about creating an Image Sequence.