I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse. By 1800, they are almost non-existant. Many B/M/D events either weren't recorded, or haven't survived.
For Henry Webb B 1804 +/- 5 yrs, somewhere in Ireland, with a father forenamed Robert, I found 1 record in 1800, Co. Dublin. However, the mother's forename doesn't begin with Mar(y). This MAY be for your ancestor as record keeping at this date was sloppy; many parents' names were simply not recorded.
Hope this helps. It's the only surviving record which comes close. Best of luck and happy holidays.