Searching for your Wickham ancestry in Co. Wexford will not be easy at present as there are very few Church of Ireland registers (i.e. Anglican and equiv. to Church of England) online for this county at present and your line was born well before the start of civil BMD records. As far as I know no major website have any of the Kilmore registers available. Kilmore was part of the parish union of Tomhaggard in Ferns diocese and according to the RCBL Parish list the available records include baptisms back to 1809, marriages back to 1813 and death/burials back to 1809, the historic registers are held by the Representative Church Body Library in County Dublin, and also on Film in the National Archives of Ireland.
There is a project underway to transcribe Church of Ireland registers run by the Representative Church Body Library who hold many of the historic registers, but this is going to take quite a while to complete - see the Anglican Record Project at present only partial records of two Co. Wexford parishesno mention yet of Kilmore/Tomhaggard.
The Tithe Applotment records, dated about 1833, do not include any Wickham entries for Kilmore parish, but that could mean they didn't have eligible agricultural property at that time, or had moved away. A general Co. Wexford search for a John Wickham in these records shows 8 matches, although it appears two of these are duplicates - none of these are in Kilmore or the other parishes in the Church of Ireland Union.
The later Griffith's Valuation records, 1853 for this area, dont include any in the parish either. Looking at surname distribution for Wickhams in the mid 1800s shows it was not a common surname, only about 66 properties in all of Ireland, the majority, 42, in parts of Co. Wexford.