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:
https://www.nli.ie/en/family-history-introduction.aspx
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:
https://www.nidirect.gov.uk/publications/proni-guide-church-records
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
I checked the tithe applotment records for Templepartick for 1833. There were no Harpers listed, indicating they weren’t farmers.
http://www.irishgenealogyhub.com/antrim/tithe-applotments/templepatrick-parish.php
Griffiths Valuation has a Samuel Harpur on plot 13c in KIlmakee (a labourer’s cottage) in 1862. Perhaps he is a relation?
It feels as though your farmers were labourers/weavers and they can be very hard to trace in the 1700s and early 1800s. They don’t show up in many records.
The church with the oldest records in Templepatrick is the Non Subscribing Presbyterian church whose baptisms & marriages start in 1796. There’s a copy in PRONI. You may struggle to find much on the family before 1800.
