The Richardsons are the London family that the Steele girls stay with after they have upset the Ferrarses with the revelation of Lucy’s engagement. Presumably, they are the relatives residing in Bartlett’s Building that the Steeles came to town to visit in the first place.
Miss Steele says of the Richardsons “I assure you they are very genteel people. He makes a monstrous deal of money, and they keep their own coach.”
Late in the novel, the Dashwoods run into Anne Steele at Kensington Gardens where she fills them (and the reader) in on the developments in Edward and Lucy’s engagement. Edward, and later Robert, appears to be calling at the Richardson home to sort everything out.