Hi,
Although Morningthorpe has the parish registers online, not all the relevant dates are included. Archdeacon and bishop transcripts are also now available on familysearch beta site, (not looked on the pilot). From the archdeacon transcripts: Image 20 has the burial of the wife of James Coe. Image 13 Henry's baptism, John's is image 14. James burial is image 43 and gives his age as 65 so probably born c 1720. Didn't find any marriage in Morningthorpe though.
Not found any marriages on the transcription sites, although there is an Elizabeth Tyrril marrying a James Coe 2 May 1736 on familysearch in Suffolk in Hart.... (??)
If you think they are from Norfolk, you might try trawling through the online parish registers or the AT transcripts. Not easy I admit.
Good luck kaysii