I too have found the Abstract of Thomas Vanatta's Will on the Sussex Co, NJ webpage.I do not know the Vanatta's full connection to the Lamberson's. I am researching Lawrence Lamberson who married Christina Fite.They left NJ and about 1789 and moved to Bedford Co, PA.They left there about 1805 and went to Smith Co, TN.You mention it was Bridget and Lawrence but it definitly was Christina and Lawrence who migrated to TN. A Bedford Co, PA Deed clearly states that they bought land in TN and Christina (living in TN) filed for a widows pention based on Lawrence's NJ Revolutionary War service.Both Lawrence Lamberson/Lambertson's were living when Christina's father died in 1782, Sussex Co, NJ.Both men are mentioned in his will.I am trying to establish if the two Lawrence's were related (ie: father/son, uncle/nephew, etc)
I find it curious that your Peter Vanatta also went to Smith Co, TN. After the Rev. War the Fite brothers (Leonard and John) left NJ and first settled for about 9 and 14 years in Lincoln Co, NC and then moved to Smith Co, TN. Being among the first settlers in that region.Shortly after the Fites arrived, Lawrence and Christina Fite Lamberson followed her brothers there.The Vanatta's, Fite's, and Lamberson's seem to have followed similar paths.
You are correct that Lawrence and Christina had a son Peter.
I will check my files to see if I have anything on the Vanatta family that could help you. Thanks for your help.