After further investigation this appears to be Vimeo blocking the requests.
We have two sites (one where Vimeo videos work and one where they don’t). When testing the retrieval of content via PHP we get “HTTP request failed! HTTP/1.1 403 Forbidden” on the faulty site.
Anyone else reading this thread can run the same test with this script:
This will identify if Vimeo is blocked on your hosting server – if it is you should contact your hosting provider and/or Vimeo to get the IP block removed.
Not a WordPress problem per se, but hopefully this diagnostic test helps someone else in the future.