No. Diana's mother remarried a Shand-Kidd. If there is a link, it is only through her mother by marriage.
There is a paternal link via marriage as well but waaaaay back- Henry Cavendish*, 2nd Duke of Newcastle (1630) had several daughters, one of which is related to Camilla:
Catherine, who married Thomas Thuton 6th Earl of Thanet (Camilla's line)
Her sister Arabella married Charles Spencer, 3rd Earl of Sunderland, whose second wife was Lady Anne Churchill. Charles and Anne are the ancestors to the Earls Spencer as well as the Dukes of Marlbourough and thus Diana through her father's line.
Incidentally, Henry is the common ancestor of Charles and Camilla as well (Charles is related to Margaret, another daughter through the QM).