The question is being flagged as off topic for violating one of the prohibited questions rules, namely "unauthorized use or misuse of IT systems".

I will explain why this is not about circumventing policy:

If the IT administrator blocks all streaming video at the firewall, we cannot and will not try to work around it. We overnight a DVD to those users. Could we ask the IT administrator to unblock only video from our site? Sure. But we don't have time to do that nor wait for it to happen. We need to get the users trained ASAP.

If the IT administrator blocks Vimeo, we want to serve the video from our site. The moderator that flagged the question suggested this approach himself:

Our server will download the appropriate version of the video from Vimeo on the fly (maybe cache it for an hour) and serve it to the client. The other resources necessary to play the video (JS, JSON) are permanently stored on our server, and are modified to tell the client to request the video from our servers.

This is not about circumventing policy. This is a creative solution to allow us to keep only one Online Video Network solution and almost always be able to deliver the video to our end-users. We have used multiple OVNs in the past, such as JWPlatfom and Kaltura, along with Vimeo, and the application would try all three of them before giving up, but there are situations in which they are all blocked specifically (as opposed to blocking any video streaming). We have now decided to use only one OVN and fallback to our server to serve the videos.

Why don't we straight host it all ourselves? Because aside from stored, online video needs to be transcoded (multiple formats need to be stored + cost/time to transcode), served (bandwidth costs), tracked (we cannot come close to building the analytics infrastructure OVNs have) and CDN'ed (which we can do but it comes bundled more cost effectively in an OVN service). Most importantly, an OVN is the most cost effective way to use these services together.

I hereby ask for suggestions on how to improve the question so it can be re-opened.