Yes, Phoenix Jackson's journey is successful. Despite a few mishaps along the way, she makes the physical journey. It is a familiar and arduous trip, but she perseveres and there is satisfaction for her in making this journey. The day she can no longer make this sojourn is probably the beginning of the end for her, as she will have lost her purpose in life.
From an existential point of view, Phoenix's journey is also successful. For, she has, like her symbolic nomenclature, risen time and time again to the needs of her grandson for whom she lives. The trek to the clinic, thus, gives her old life meaning and purpose.
Strictly speaking, yes, Phoenix's journey in "A Worn Path" is successful. She gets the medicine she needs for the boy, as well as a coin or two to buy him a little present.
The idea in the story, however, is that the journey Phoenix must take time and time again is a part of a no-win situation. The path is worn because the boy's illness is incurable, so Phoenix must take the journey repeatedly. The path will continue to get more wear. By extension, paths like this one are worn throughout the South and throughout the world--anywhere human beings are at the mercy of indifferent social forces.
Notice that no one is really out to get Phoenix. But no one really has her best interests in mind, either. No one does her any physical harm. But no one really goes out of his/her way to help her. The man on the path might be a bit depraved, but he doesn't physically hurt Phoenix. Phoenix gets the medicine through charity, but grudgingly and only after she is at least slightly humiliated.
Phoenix's journey is as successful as it can be in the South while it's recovering from the Civil War, but only in part and only temporarily and provisionally.