Hi Kaitlin,
I'm sorry to say that there isn't a self-hosted version of Atlassian University available at this time so unfortunately that wouldn't be possible and therefore I would recommend that ou try to figure out a way around the corporate network limitations, for example even using a phone to watch the videos might be a solution to workaround the limitations of your network or you could use a broser addon to download the YouTube videos as that would also make them available inside your network.
All the best,
John
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