I agree "out of date" isn't quite appropriate here, as it seems to suggest that the vaccination doses themselves are out of date (i.e. "past their use-by date"). I'd probably say "Are you behind with your vaccinations?".
Let's suppose that we are talking to a non-medical person to find out if his vaccinations are up-to-date, and not discussing the language used to medical people to find if batches of the vaccine are past their sell-by date.
I think I'd say: are your vaccinations (or jabs) up-to-date? or are you up-to-date with your vaccinations? The formula suggests that these are vaccinations which need to be renewed occasionally, like tetanus jabs.
I don't think I would ever use, or have ever heard, an antonym for up-to-date used in this context.
I agree: Just end the question with ...up to date? Or: ...up to date, or not? We don't call them "jabs," by the way; here, they're shots. (Except for a couple of oral and nasal vaccines, used mainly for little kids.)