Closing a registration is done by editing it and setting the status to "closed". This can happen either via Special:EditRegistration or the "update event registration" API endpoint. Currently, both the special page and the API endpoint can only be used on the wiki where the event page is, as per T307358. While this limitation makes sense when changing the event page, it should still be possible to change other fields from other wikis via the API. This is currently not possible because the "update event registration" endpoint uses PUT, which requires a full representation of the resource. A possible solution could be to create a PATCH endpoint, where you could specify only the attributes of the resource that you want to change (in this case, the status). However, I'm not sure if the MW REST API framework supports PATCH; additionally, writing such an endpoint wouldn't be trivial (it also shouldn't be complex, but I don't think it's worth doing that at this time).