You have put this enquiry on the board for Armagh but how certain are you that’s where Thomas was born?
If Thomas’s father James was a farmer, he should be in the tithe applotment records. These were compiled across Ireland between 1825 – 1838. There were a number of James Early/Earley farmers across the counties what is now Northern Ireland. Nearly all lived in Co. Tyrone. None lived in Co. Armagh.
Early, James-T: Ardarrow Y: 1826-Termonamongan-Tyrone
Early, James-T: Errigle Y: 1833-Errigal Keerogue-Tyrone
Early, James-T: Gort Y: 1833-Errigal Keerogue-Tyrone
Early, James-T: Tenements Y: 1833-Errigal Keerogue-Tyrone
Early, James-Townland: Drumard Year: 1825-Ballynascreen-Derry
Early, James, Jr.-T: Gort Y: 1833-Errigal Keerogue-Tyrone
Early, James, Sr.-T: Gort Y: 1833-Errigal Keerogue-Tyrone
Early, Jas.-T: Dernenagh Y: 1829-Pomeroy-Tyrone
Early, Jas.-T: Legnagulloglagh Y: 1826-Leckpatrick-Tyrone
Earley, James-- Townland: Drumkee Year: 1825-Killyman-Tyrone
Earley, James-- Townland: Drummuck Year: 1825-Killyman-Tyrone
Birth registration didn’t start in Ireland till 1864. Prior to that you need to rely on church records, where they exist. So you would need to search the baptism records for the parishes listed above. 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 of Tyrone, there were 180 people named Earl(e)y. The vast majority were RC but there were some Church of Ireland and Presbyterians too.