There are several reasons the author might have set the story during December and the Christmas season. First and foremost is that Christmas brings to mind love and sacrificial gift-giving, and both of these are strong factors in what motivates Old Phoenix to make the long journey into town to get the medicine for her grandson's throat.
Phoenix wants to do what she can to ease her grandson's pain and suffering, so she is willing to endure the physical travails of trudging along the worn path on this "bright frozen (December) day in the early morning" despite her advanced age and her own physical ailments and limitations. This journey, perhaps, also brings to mind the long journey the wise men made to bring gifts to Jesus. They, too, gave gifts from their hearts to help meet the baby's needs.
Along the way, the white hunter condescendingly suggests to Phoenix that perhaps she is "going to town to see Santa Claus." This adds to the Christmas setting, as does the nurse giving Old Phoenix a nickel from her own purse because it is the Christmas season.
This nickel, in addition to the nickel Phoenix picked up earlier when the hunter dropped it, enables her to buy her grandson a special gift for Christmas. She announces to the nurse and the attendant at the desk:
"I going to the store and buy my child a little windmill they sells, made out of paper. He going to find it hard to believe there such a thing in the world."
Finally, with Phoenix being so old, she can be said to be in the winter, or "December," of her life. Perhaps this was also part of the author's thinking in setting the story in December.