Where I live in Hungary, they have started some kind of registration with some technical problems (lack of confirmation e-mails in some cases), but currently they are looking for other sources of vaccines with less-known background and it is unclear which vaccine they would like to use for a particular person later on.
However, approved vaccines of the EU come really slowly, so it is also unknown for us - with lower risk - when would be available even now, when the registered user base here is relatively small (I think the relative number of vaccines available are similar in EU countries,