I believe I understand what happened here. After talking to Vimeo support, the most likely explanation is that we ran into a rate limit on their oEmbed endpoint. They mentioned:
One behavior our system checks for is the number of incoming requests made by a single IP; if a server were to make thousands of requests to Vimeo within only a minute or two, it is extremely likely the server’s IP would be banned.
This didn’t make sense to me until I realised I’d been running a large rebake this weekend, which potentially could have sent many requests their way in a short amount of time (we’re a graphics community and feature many videos). Their bans last for 7 days, which means ours will be lifted in two days.
I have good faith that this won’t be a problem again, UNTIL we need to do another rebake that is. At that point I’ll have to figure out how to rebake all posts that don’t contain any vimeo links at full speed, and rate-limit the others. Sounds doable.