] The Vyne is one of my favorite "JA places"- although, really the connection to her directly is a bit tenuous.
One of the sons of Mrs. Lefroy (JA's friend, neighbour, mentor and distant cousin), John Henry George Lefroy (known as George) married one Sophia Cottrell in 1806. Sophia was descended from the Shutes of the Vyne and was therefore a cousin of the Chute family.
At this point John Henry George took over the rectory of Ashe from his father. He was then the neighbouring rector to JA's eldest brother, James who, similarly, had taken over the rectory at Steventon from his/JA's father, George Austen.
This connection was re-enforced when JA's eldest niece, Anna Austen (James' daughter) then became Sophia's sister (sister-in-law) when she married Benjamin Lefroy, John Henry George's brother, in 1814.