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.
The question comes to this: what is the difference between self hosting and relaying the video thru our server? The video is coming from our servers either way, but in the opinion of the moderator who locked the question, whether the video sits on our server OR is being relayed thru our server determines whether policy is or isn't being circumvented. Ok then. How about this? My application sits on a server in the Google Cloud. The videos are stored on S3. S3 is blocked by the client's firewall. We only serve the video thru our application server. Is policy being circumvented in this case?
BTW: all of this is OUR content behind password protection in our application. Additionally, we cannot possibly contact IT for all these facilities all over the world - some of them barely speak english. Others have no dedicated staff. Others are buried in bureaucracy. Most importantly, they decided to block Vimeo, which is understandable, and they can't unblock only OUR videos at Vimeo.
I hereby ask for suggestions on how to improve the question so it can be re-opened.
This is a creative solution. This is also a narrow edge case, narrowed by your focus on what you perceive as a problem, with a solution that is confined to your highly-filtered view of how to solve it. That is two additional problems with this question. A question too narrow may not be of benefit to many others, and may not a good candidate for the forum. A question that requires a solution is confined to very narrow requirements or a highly-filtered view of how to solve it by the person asking for help limits the abilities of others to address what may be the real problem.