Statutory registration of births didn’t start in Ireland till 1864 so you won’t get a birth certificate for either William or Elizabeth. However you might find their baptisms, but to do that you need to know their denomination and where they attended church.
A Presbyterian living in Staffordstown would likely attend church in Randalstown. There are 3 Presbyterian churches there. The baptisms for the Old Pres Church start in 1853, marriages 1845; for the 1st Pres church I don’t know when their baptisms start, but marriages start in 1845 and for the 2nd Pres Church their baptisms start in 1850 and marriages in 1845. (The churches are all much older than that but evidently earlier records are lost). So you might not find William’s baptism, but you might find siblings or his parents’ marriage.
If Elizabeth was also Presbyterian then Donemana Presbyterian church baptisms start in 1856 and marriages 1845. So you might scrape in with her baptism.
None of those records are on-line. However they have been copied and can be seen in PRONI, Belfast.