Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Looking at the 1901 census for Armagh, the vast majority of the 361 Hewitts were Church of Ireland (ie Anglican). There were a handful of Presbyterians , Brethren & Quakers but 85% were C of I. Seagoe Church of Ireland records start in 1672 (with some gaps). The earliest Presbyterian records appear to date from 1839, Quaker records go back to the 1700s and Brethren don’t keep records. Hopefully your family were Church of Ireland. If so you may find them in the Seagoe records in PRONI.