According to those who hold that Christianity is avodah zarah, may Jews do business with Christians within three days of Sunday (or if you want, within 3 days of Dec. 25 or Easter) in light of the prohibition (Rambam Avodah Kochavim 9:1). If so, why?
Tosafot in Avoda Zara, 2a s.v. אסור answer this question. (By the way, not every one agrees to the "three days" thing Avodah Zara 7b.)
A: because of איבה, i.e. if Jews never did business with Christians before their holidays this would cause undue hatred of Jews. However, the Tosafot reject this, because there is no hatred, because the Jew could just say that he didn't need to buy from him that day.
B: (not really relevant, because you were asking your question not according to this answer) Christians are not idol worshipers.
C: (Rabbeinu Tam) It is only אסור to sell things to idolaters that will be used as an offering, and it is completely מותר to buy from them.
According to the Mishneh Torah, Avodah Zara ve-Chukot ha-Goyim 9:4, Christians are Ovdei Avodah Zarah( idolaters), Sunday is their Yom Eidam( holyday, festival), and one is forbidden to do business with them, on Thursday till Sunday of every week in the Land of Israel, but only on Sunday itself everywhere else. Same goes for their other holidays.