Do you have years of birth, even approximate, for any of your Cunningham family ?
Full Civil records started in Ireland in 1864, so for anything before that you would need to check for church records. In addition to the civil records from 1864 onwards, non-Catholic marriages were registered from 1845 - which might assist your search if for example your Cunningham family were Church of Ireland.
The Catholic parish for Labasheeda is 'Kilmurry-McMahon', sometimes listed as 'Kilmurry McMahon & Killofin', Killofin being the name of the civil parish which covered the town of Labasheeda. Baptism records for this RC parish seem to go back to about 1845, marriages to 1837. Like many Co. Clare parish records these dont appear to be available online.
There are just three Cunningham households listed as holding property in Killofin civil parish on Griffith's Valuation (a taxation survey) - Margaret with a house but no land in Cloonarass townland, John and Thomas also with houses and no land in Lakyle north townland.
see :
Cunningham properties, Killofin Civil ParishThe Griffith valuation records for this part of Ireland are dated 1855.
Shane