Vatican City is actually NOT part of the European Union — they won’t let them in!
The Holy See, which has Vatican City as its sovereign territory, is not eligible to join the European Union. It does not meet the EU's Copenhagen criteria because it is not a free market democracy. The European Parliament also does not agree with the Holy See's view on homosexuality.
The Holy See is viewed the same as a sovereign state, having a centralized government, called the Roman Curia, but it's a theocracy, not a democracy. A theocracy is a system of government in which somebody, usually a priest, rules in the name of a god. In this case, the Pope rules in the name of the god of the Catholic Church. Diplomatically, the Holy See acts and speaks for the whole Catholic Church.
Because it is so small and surrounded by an EU state, it is intrinsically linked to the EU, even if it's not eligible to be a part of it. It has an open border to the European Union and uses the euro as its only currency. The Vatican is exempt from taxes and duties. The very small amount of exported goods from the Vatican is therefore duty free.
The European Union granted the Holy See permission to mint one million euros a year and an additional 300,000 on special occasions.