Whenever a new country becomes a member of the Eurozone, they convert all their local currency into the Euro. What I don't understand is where does this huge new volume of Euros come from overnight? Does the European Central Bank simply print the extra money? If so, why doesn't this cause significant inflation?
Yes the European Cenral Bank issues the extra quantity of money (as for the actual printing there are factories that print euros in many member countries of the EU, under the control of the ECB).
Regarding why this does not cause inflation, it is because alongside the additional quantity of money comes the additional quantity of output of the new member, and the additional transactions related to this additional output that the additional quantity of money is called upon to facilitate.
Use the traditional "quantity theory of money"
$$PQ = VM$$
$M$ increases but $Q$ also increases, so $P$ does not have to increase.
Of course it is much more complex and complicated and gradual in a real world situation, but the fundamental idea is the above.