Symbolism, Imagery, Allegory
There is a production note in Sam Shepard's Seven Plays regarding the coyotes of True West. It pretty much sums up their symbolic role in this play:
The Coyote of Southern California has a distinct yapping, dog-like bark, similar to a Hyena. This yapping grows more intense and maniacal as the pack grows in numbers […]. The sense of growing frenzy in the pack should be felt in the background, particularly in Scenes 7 and 8 (Shepard, Seven Plays 4).
The phrase "growing frenzy" says it all. The coyotes loom on the outside, like some impending message of doom. As the brothers transform and descend into chaos, the coyotes are there to show us that Austin and Lee are becoming more and more primal—more animal-like. The coyotes symbolize the animal in all of us. They are also a symbol of the West that once existed—the untamed land ruled by nature and violence.
Toward the end of the play, Shepard starts to give the brothers more animal-like qualities. Lee "begins to circle AUSTIN in a slow, predatory way" (2.8.270-271). In the end, as the brothers stare each other down, waiting to see what happens next, "a single coyote" can be heard in the distance. The Old West has invaded the New West. The brothers are truly animals now, and perhaps even more important, they are the same, single animal.