To make the question easier, suppose that the recipient of the Christmas gift is a devout Christian who actually celebrates for religious reasons.
Potentially there are several issues with buying a Christmas gift for such a person:
- It is forbidden to benefit from anything related to idolatry, because of “Do not bring the abomination to your house” and “Let nothing from the Cherem cling to your hand.” (See Rambam, Avodah Zarah 7:2)
- One is prohibited to do business with idolaters near their festivals (AZ 2a). There are two reasons brought (ibid. 6a, as explained by Rashi):
- The non-Jew will thank his gods (a violation on the Jew’s part of “the name of other gods...may not be heard on your account”)
- The non-Jew will use the object for his idolatrous service (a violation on the Jew’s part of Lifnei Iveir, causing another to sin)
Christianity, according to some is considered Shituf, serving Hashem along with an idol, and is only prohibited for Jews, not non-Jews. I’d imagine that the Lifnei Iveir issue would therefore not apply, and I would imagine that “the name of other gods...may not be heard on your account” would apply.
Would there be an additional prohibition because of possession of idolatrous items? On the one hand, Christianity is idolatry for Jews, and so it might be prohibited. On the other hand, it’s not idolatry for its recipient, and since it’s intended for that Christian, perhaps it would be permitted.