Christine - You won't find a birth registration for the children because English Birth Registrations did not begin until 1837.
I have stumbled on a brick wall that I cant seem to get past and wonder if you can help point me in the right direction.My Gt gt gt grandfather John reay( 1790c) and his wife elizabeth ? nee Barker(1796c born in whitby) are named as parents of Edward REAY in Upleatham yorkshire in 1819 on the Baptism certificate( IGI).they then went on to have John and William 1926 [sic] Hannah 1831, Robert 1832, betsy ann 1836 in Shinclffe durham.
IN 1841 census Elizabeth reay is listed as independant, and there is no sign of John. I cant find birth certificates for the children, nor can i find any marriage for Elizabeth and John or any evidence he existed except for the baptism certificate. In 1851 elizabeth is listed as Murray and a widow in Hartlepool, but I have no idea who or when she remarried or when John ( supposibly died). Where can i search next??
However you can search for Church and Parish records of baptisms and other vital records. Because I am not familiar with Durham research, I began with a quick Google search for Durham Church Records. Of course you can use any Search Engine you prefer such as Bing or Yahoo.
Durham Church Records are online at http://www.durhamrecordsonline.com/ and a very quick look under baptisms brought up the baptism record index for Betsy Ann, Hannah and Robert all baptised in Shincliffe. There is also a marriage in 1819 for John Rea [sic]. You may wish to consult this to see if it is the right couple.
I'd also check Durham University's databases online. I found that website with a Google search and notice they have Probate records.
You mentioned you checked the IGI but a better course of action might be FamilySearch collections that they have been putting online. I did a very cursory search and found the marriage of John and Elizabeth in the extracted marriages for York. They married 18 Aug 1816 in Loftus [Source: England, Marriages, 1538–1973]. So now you have other places to search. If I were you I'd start a timeline of findings so that you can get a sense of what location(s) you might find the family in.
Their son Edward is found in baptismal records as being baptised 01 November 1816 (not 1819 as you stated) in Loftus so it looks like they married just in time...