I am familiar with two families of Ansteys
There is a camp site business in Twillingate that was owned by the Ansteys. It is called Peyton Woods Campgrounds.
There is also an auto / scrap metal recycling business in St. John’s, owned by Bob Anstey. This is called A1 Automotive. He also runs Newco Recyling Ltd. at the same location.
I don’t have any ancestry information on this family but these connections may help ?