Contrary to what you might expect, there are no pilgrims at Tinker Creek—or at least, not the big-hat-and-buckle-shoes kind. Instead, Dillard's using the word more generally, to refer to "a person who journeys to a sacred place for religious reasons." Tinker Creek is her sacred place, and she's there to search for God. Dillard's looking at creation in an attempt to see the creator—Dillard is the pilgrim.
How, you might be wondering, does someone attempt to see the creator? After all, that's a pretty tall order. Dillard tries two approaches based on the teachings of ancient Christian mystics: the via positiva and the via negativa. In the first half of the book, she assumes that God is omniscient and good, while in the second, she assumes God is cryptic and exempt from human morals. Either way, she's out there pilgriming around, looking at muskrats.