Will people born after the Rapture continue the Church?
There is no Church after the rapture. There are believers, but the "church" is a unique thing: The Bride of Christ – those believers from Pentecost to the Rapture.
The people who come to faith after the rapture are called Tribulation saints. There may be a gap between the rapture of the church and the start of Tribulation, but it may not be very long. Nevertheless, we do read of new faith beginning with the start of Tribulation (Revelation Chapter 7 talks of a great revival movement).
They are simply “believers”, as the “church” is only for our period of history.