Figured it out on my end. It was due to the Permissions-Policy being set by Nginx in the header. In my instance of Nginx, it was originally set to this:
add_header Permissions-Policy "geolocation=(),midi=(),sync-xhr=(),microphone=(),camera=(),magnetometer=(),gyroscope=(),fullscreen=(self),payment=()";
The culprit in this case was fullscreen=(self) -- it was telling Chrome that unless the code originated from the site, it shouldn't allow full screen. As Vimeo's iframe is being loaded from player.vimeo.com, Chrome saw that as a third party and wouldn't allow it. Removing that from the Permissions-Policy so it looked like this:
add_header Permissions-Policy "geolocation=(),midi=(),sync-xhr=(),microphone=(),camera=(),magnetometer=(),gyroscope=(),payment=()";
Resolved the problem. The button is showing fine now.
For those using Apache, it'd probably look like this:
Header always set Permissions-Policy "geolocation=(),midi=(),sync-xhr=(),microphone=(),camera=(),magnetometer=(),gyroscope=(),fullscreen=(self),payment=()"
The same principle applies, just remove fullscreen=(self).
You might also see it mentioned as Feature Policy, same thing, it's just called Permissions Policy now.
You can learn more about Permissions Policy here: https://github.com/w3c/webappsec-permissions-policy/blob/main/permissions-policy-explainer.md