The EU has paid AstraZeneca to build manufacturing capacity within EU borders, but a big part of that capacity is used to supply the UK. AstraZeneca has also quite smartly been extremely slow to file the necessary paperwork to make those plants EU compliant so that even with export controls, the capacity of those plants cannot be used by Europe.
Edit: The comment originally claimed that EU made vaccines were exported to the USA, this is actually not the case
Edit: The comment originally claimed that EU made vaccines were exported to the USA, this is actually not the case