Statutory registration of births in Ireland didn’t start till 1864, and marriages weren’t registered till 1845, so for this family you won’t get marriage or birth certificates. You may find a record of their baptisms and marriage in church records but you need to know their denomination to do that.
The local Church of Ireland has records from 1783, which include some Presbyterian baptisms for the years 1822-1831.
Clare Presbyterian has records from 1822, Cremore 1831, Tandragee 1835 & Tyrone’s Ditches from 1793. The RC records start in 1843.
The RC records are on-line on the nli site, but for the others you will need to go to PRONI (the public record office) in Belfast and look them up there.
If you think William & Elizabeth might have been alive in 1864 (when death registration started) you could also search for their death certificates. Ideally you need to know his townland or occupation to be sure of finding the right one. Check Witten as well as Whitton as both spellings were common.
Griffiths Valuation for 1863 has a William Whitton farming in Mullaghglass and another William farming in Tullyhugh. Deaths around Tandragee should be registered in Banbridge.
I see a death for Wm Whitten of Tullyhugh on 30.11.1890, aged 75. He was a widower. Informant was his son in law Thomas Annett.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1890/06091/4738140.pdf
An Eliza Whitten died in 1871 aged 50. That certificate is not on-line free yet, so if you want to view it, you’ll need to pay. You can view the original certificate on-line on the GRONI website, using the “search registrations” option:
https://geni.nidirect.gov.uk
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate.
The Will of William Whitten late of Tullyhue County Armagh Farmer who died 30 November 1890 at same place was proved at Armagh by Bessie Annett (wife of Thomas Annett) of Tullyhue one of the Executors. The will is on-line on the PRONI website. In it, he left everything to his daughter Bessie Annett. This looks to be her in 1901:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Tandragee_Rural/Tullyhugh/1021156/
Possibly her husband living nearby in 1901:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Tandragee_Rural/Tullyhugh/1021136/
1911:
http://www.census.nationalarchives.ie/pages/1911/Armagh/Tanderagee/Tullyhugh/332482/
The above family was Church of Ireland. I cannot say that it is your family merely that the information fits with what you have given so far. You would need to search the baptism records to see if you can find Emma and perhaps siblings.
Here’s another William Whitten born c 1832, in Mullaghglass. Again Church of Ireland.
http://www.census.nationalarchives.ie/pages/1901/Armagh/Tandragee_Rural/Mullaghglass/1021105/