Q. As the holiday season approaches is there a problem with sending a gift to customers or business acquaintances?
A. The Talmud (Avoda Zarah 20a) instructs that there are different interpretations to the biblical proscription of "Lo Techonem" (Devorim 7:2) one is; "not to give them free gifts". Who exactly qualifies as "them" and what is a "free gift", is open to Halachik discussion. Shulchan Oruch (Y.D. 151,11) rules: It is prohibited to give a free gift to an idol worshiper you do not know. Taz (ibid. 8) explains that if you know him, it is not considered a "free gift", as you would expect a favor in return. As such, gifts to customers or business acquaintances, even if they were to qualify as idol worshipers are not prohibited.
However, based on a the ban of promoting and encouraging idol worship mentioned in the Talmud (beginning of Avoda Zarah), Shulchan Oruch (Y.D. 148, 5) rules that it is prohibited to send a gift to an idolater on the day of his festivity, unless it is known that the recipient does not believe or worship idol deities. As mentioned in a prior answer, there are different opinions as to which religions are considered idolatry.
Horav Shlomo Miller's Shlit"a opinion is indeed to avoid sending gifts that would reach a Gentile on the day of the celebration. It is better to shun the prior day also.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit"a