Edit: This feature is now supported by Vimeo (in beta) API3. You can request access at https://vimeo.com/help/contact
The information below is still relevant for the advanced API.
This feature is not reliably supported by the extended api.
The main problem is that when downloading on the client side there is no way to tell the download server what to do after the download is complete. In addition, there is no technology on the client side to find out if the download is complete.
You might be able to spoof and load the user through an iframe, and the primary page checks for pieces of packets every couple of seconds. I have not tried this, at the moment it is just a theory. If you try to perform this hack, Vimeo would like to know more through its contact page https://vimeo.com/help/contact
The feature you are discussing is on the roadmap for the next version of the API, and an early imperfect version was available in their Hackday API. Perhaps it would be better to stop a bit and wait for the official function.
NOTE. This observation does NOT apply to server-side downloads. Since the entire download process is processed using code on the server, it is very easy to tell when the download is complete.
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