Is it permissible to speak and move (such as taking a child to the bathroom) after any two rak'ahs of the night vigil prayer and uttering the salaam?

There is no objection to interspersing the night vigil (Qiyam al-Layl) with some actions and activities. Indeed, Ibn Abbas spent the night at the house of his aunt Maymunah. The Prophet (peace and blessings of Allah be upon him) prayed 'Isha, then prayed four rak'ahs, then slept. Then he got up and prayed five rak'ahs, then prayed two rak'ahs, then slept.