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 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:
In this case, not knowing exactly where the family came from will make the search very difficult. There are a lot of churches in Tyrone and the other 2 possible counties. Not all of them have records for the early 1800s either so that won’t help.
You haven’t said what denomination the family was but from the names I’d guess Church of Ireland (Anglican) or Presbyterian. The Church of Ireland is slowly putting all it’s records on-line. The project has a couple of years to go, but eventually you should be able to search them all on the irishgenealogy database. You might get lucky from that.
The usual advice in this situation is to try and exhaust all possible sources in Canada in the hope that you find a clue about where they originated. If you can get a parish or a townland it becomes a lot easier. Check all death and marriage certificates, wills, military records, obituaries, gravestones etc in the hope that one of them will mention a place in Ireland.
The name Thompson/Thomson is fairly common in Tyrone. In the 1901 census of the county there were 557, of which 14 were named Thomas. It would have been equally common in the 1830s so you will need to take care to find the right family. I’d suggest you check out Thomas’s sister’s marriage certificate in Toronto to see precisely what it contains. Not only should it confirm her parents names but you might get more than just Tyrone. Incidentally, since the information on her birth certificate would have been first hand, you would expect it to be more accurate than information handed down second hand. So for that reason, if it does say Tyrone, I’d probably focus on that county, rather than Donegal or Armagh. Still a lot of records to go through without more detail.
One question is what was John Thompson’s occupation (perhaps recorded on John’s marriage or death certificate)? If a farmer, and alive in the late 1820s/early 1830s then he’s likely to have been listed in the tithe applotment records. Here are all the John Thompsons listed there for Tyrone. Perhaps your John is one of them.
Thompson, Jno.-T: Corlea Y: 1826-Kilskeery-Tyrone
Thompson, Jno., Esq.-Tl: Ballymurphy Yr: 1826-Arboe-Derry and Tyrone
Thompson, Jno., Esq.-Tl: Ballymurphy Yr: 1826-Arboe-Tyrone
Thompson, Jno., Esq.-Tl: Turkvallen Yr: 1826-Arboe-Derry and Tyrone
Thompson, Jno., Esq.-Tl: Turkvallen Yr: 1826-Arboe-Tyrone
Thompson, John-T: Bearagh Y: 1826-Clogherny-Tyrone
Thompson, John-T: Coagh Gardens Y: 1827-Tamlaght-Derry & Tyrone
Thompson, John-T: Coagh Gardens Y: 1827-Tamlaght-Tyrone
Thompson, John-T: Coagh Y: 1827-Tamlaght-Derry & Tyrone
Thompson, John-T: Coagh Y: 1827-Tamlaght-Tyrone
Thompson, John-T: Corlea Y: 1826-Kilskeery-Tyrone
Thompson, John-T: Eskermore Y: 1826-Clogherny-Tyrone
Thompson, John-T: Lisnahull Y: 1826-Donaghmore-Tyrone
Thompson, John-Tl: Aghasessy Yr: 1834-Ardstraw-Tyrone
Thompson, John-Tl: And. Torrens's Meadows Yr: 1825-Artrea-Derry and Tyrone
Thompson, John-Tl: Ballimaquiggan Yr: 1825-Artrea-Derry and Tyrone
Thompson, John-Tl: Town and Lands of Caledon Yr : 1837-Aghaloo-Tyrone
Thompson, John-Townland: Cliffany Year: 1834-Aghavea-Tyrone and Fermanagh
Thompson, John-Townland: Coolnakelly Year: 1834-Aghavea-Tyrone
Thompson, John-Townland: Crohan Year: 1833-Aghalurcher-Tyrone
Thompson, John-Townland: Slushhill Year: 1833-Aghalurcher-Tyrone and Fermanagh
Thompson, John-Townland: Tullyreagh Year: 1834-Aghavea-Tyrone
Thomson, Jno.-Tl: Ballindrum Yr: 1825-Artrea-Derry and Tyrone
Thomson, Jno.-Tl: Ballinuey Yr: 1825-Artrea-Derry and Tyrone
Thomson, John-T: Anaghmore Y: 1827-Clonoe-Tyrone
Thomson, John-T: Aughagala Y: 1827-Clonoe-Tyrone
Note: Several of the entries give two counties. This is because the information has been compiled by parish, and some parishes straddle the borders of 2 counties.
Thompson’s not a native Irish name. That, plus the fact the family lived in Ulster, would point to them being descendants of folk who settled in the area in the 1600s as part of the Plantation of Ireland. Probably from Scotland. Today here in Ireland we call them Ulster-Scots. In North America they are often known as Scotch-Irish.