Not strictly, no. Even as EU members, health policy remains a national responsibility. (Just like the colour of the passports.
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It just happens that the EU members decided to go for joint procurement of the vaccines, reasoning that with common open borders there was no point in a nation by nation approach. But it has backfired rather, since the joint procurement seems to have been done too much in a standard procedural way, presumably to avoid complaints from any of the member states about shortcuts in the process. If the UK had still been a member we could - and perhaps might- have opted out of such an initiative, on the basis that we are an island and not in the Schengen system.
But one could argue that the EU is at a natural disadvantage compared to a nation state on something like this, as its central institutions don't (yet) have the authority, on health policy, to set a course of action, overrule any complaints and just get on with it.