The identity of the audience is unclear, because it is never mentioned nor acknowledged. While this is fairly common for literature, the subject must be clarified in order to answer the question in a satisfactory way. 

The best term I can apply here is the "fourth wall", which is normally used in theater and television to refer to an imaginary wall between the performers and the audience, in contrast to the three walls that form the set in which the performers act. Hypothetically, a play could be conducted with no audience, and thus the performers would be their own audience. By contrast, a book that is not being read has no audience, and its "performance" ceases to exist when the reader stops reading. 

Thus, without getting too deep into the philosophy and metaphysics of literature, it is the "default setting" for a book not to acknowledge its audience. This may be because literature is naturally derived from the "storyteller" method of communication - a one-way delivering of information from one person to one person, or a small number of people. It is assumed that we become part of the very small community that consists of the narrator and their audience, and our role in that community is to follow the narrator unless instructed otherwise.

In "There Will Come Soft Rains", the narrator is omniscient, capable of seeing all parts of the house at all times, and so we might imagine that we are illusionary observers joining the narrator; that we too are third-person and omniscient in relation to the house. We might also consider that the audience is composed of "normal", modern humans, who can appreciate the message in the story, and the irony in elements such as the poem being read (suggesting that nature doesn't care about humanity's absence).