Birth registration didn’t start in Ireland till 1864 so you won’t get birth certificates for the children. You may find their baptism though. You haven’t said what denomination the Henderson family were, but if they originated in Scotland, the most likely is Presbyterian. There are 4 Presbyterian churches in the City of Armagh. The only one with records for the period you are interested in is Armagh 1st and those have gaps. They have baptisms for 1796 to 1803 then there’s a gap till 1825. There are 3 other Presbyterian churches in the city but I am not certain that they were functioning in the early 1800s. Armagh 2nd’s records start in 1825 and it’s possible that’s when it opened.
The records for Armagh 1st are not on-line anywhere so far as I am aware. However there is a copy in PRONI (the public record office) in Belfast. A personal visit is required to view them.