There are microfilmed records for the town available for rental through a family history center run by the Church of the Latter Day Saints. The cost per rental is abt $7.50, and the microfilm will remain at your center for abt a month.
Registri dello stato civile di Delia (Caltanissetta), 1820-1915
This would be the film you need for 1836 births. Nati 1825-1846 FHL INTL Film 1438566
The other option you have is to email the town's anagrafe office, if you have some specific info about your ancestor. The office may provide you with the date of birth and the names of the parents. The email, however, should be in Italian. I don't know if the following email address will work though, but you could try it.