Searching James "John" Moore and Mary Armstrong

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.  Ideally you need to know the precise denomination and have some idea of where the person was born in order to search the correct records. Not all churches have records for that period and not all that do 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

In this case I’d guess that the Moores and Armstrongs may have been Presbyterian. (So Scottish origins). In the late 1700s there were at least 100 Presbyterian churches in Antrim but only about 6 of them have any records for that period. You could get a researcher to go through them all but statistically there’s isn’t a great chance of success. The records are mostly not on-line by the way so a personal visit to PRONI is required to view them.

Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.

The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price.  http://www.nifhs.org (Go to DNA project on the website).