No. In the GoAnimate for Schools platform, only teachers have downloading/exporting/embedding privileges, for their videos and their students’ videos. Students do not have these privileges. This is for privacy and security purposes. Students are not able to share their information without a teacher's supervision, as part of COPPA (Children's Online Privacy and Protection Act).
While this adds an extra burden for a teacher to handle the downloads/exports/embeds for many students, we can't open this feature up to students.