School Library Journal has been keeping track of different publishers' opinions on copyright and online storytimes. Each publisher has their own rules, but they all boil down to some basic guidelines:

1. First, livestreaming is covered, no problem. That's a temporary video that falls largely under the same copyright rules as normal face-to-face storytime.
2. Second, if the livestream is archived, that's fine as long as the video is not publicly available. For example, if you do a YouTube Live reading, make sure the finished video afterwards is set to "Unlisted" (not "Public"). You can post links to the video in private, school-specific platforms (such as Google Classroom) but you can't share it with the public.
3. Third, proper attribution is required. Give credit to the author, illustrator (if applicable), and publisher in the video, and provide links and/or tags in your video description.