Welcome to Ireland Reaching Out and apologies for the delay in getting back to you.
One way of looking for that connection, at least online as far as the 1920s would be to have a look at the Griffith's Valuation and track the Lamie/Lammie names that appear in it through the Cancelled/Revision Books. As far as I am aware more recent Books are only available to view in person in Belfast.
You can learn more about the Books here. I suggest you have a read of the link as they are a great source:
http://www.irish-genealogy-toolkit.com/revision-books.html
You can then move to the PRONI website to see if they have any reference to the family in their digitised online Books (the Books are not online for the Republic of Ireland).
Here is the link to the PRONI Books:
http://www.proni.gov.uk/index/search_the_archives/val12b.htm
One problem however, is that I don't know if the John Lammie listed below in the Griffith's Valuation is your John. Given that the Lammie/Lamie name is not terribly common I suggest that even if this is not your John it would be a good exercise to see where the above research leads you.
John is listed here, so this is your starting point:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam…;
As an aside, there are some Lammie family recorded on the 1911 Census:
http://www.census.nationalarchives.ie/search/results.jsp?census_year=19…
Hope this is of some help to you
Best wishes
Clare Doyle
Genealogy Support
