Hello,
I have the West Lothian Burial CD which contains interments for Linlithgow to 1974 - There are 61 entries for the name Gray - none for Carlaw. In the 1851 Census there is a William Gray aged 73 born Livingston, living at Wester Tartraven, Linlithgow with wife Mary born Linlithgow aged 67 - is this the family you are looking for? Strangely enough in the 1841 census William is aged 50, Mary is 45 and Margaret is aged 15, the address is still Wester Tartraven. Quite a jump in ages! I can't find either William or Mary in the Burial Registers for Linlithgow or Livingston. There is a marriage between William Gray/Mary Carlaw of 2nd and 21st August 1818 - the first date was probably the calling of the Banns, the 2nd date the marriage. I can't find any children born to this couple - other than the Margaret aged 15 in the 1841 census.
Sorry that I didn't find anything for you. They may have been interred somewhere else local. I did a search for William Gray deaths in West Lothian between 1851 and 1861, there were three - one in Abercorn in 1852, one in Bo'ness in 1857 and one in West Calder in 1858. The Bo'ness William had the middle name of Tassie and the West Calder one had the middle name of Simpson. I don't know if this will hep or not. Good luck ...... Meg