What is the ruling on vocalizing during the missed rak'ahs of a prayer that is ordinarily recited aloud, whether it be the 'Isha or Maghrib prayer, if the congregant caught the Imam in the last rak'ah, then stood up to complete his prayer and became an Imam?

First: Following a latecomer (masbuq) in prayer is a matter of scholarly disagreement, but the most sound opinion is that it is permissible.

Second: The answer to your question depends on the most sound opinion among scholars regarding what a latecomer (masbuq) catches of the Imam's prayer – whether it is considered the beginning of his prayer or the end of it. This is a matter of disagreement with three different views.

Based on the most sound opinion, a latecomer who catches the last rak'ah of the 'Isha prayer with the Imam should recite aloud in one of his remaining rak'ahs, as it is the second rak'ah for him. He should recite silently in the rest. However, if he recites aloud in two rak'ahs, there is no harm in that.