Is it permissible to purchase cosmetics from a store that offers discounts for Christmas, knowing that celebrating Christmas is an innovation, and is it permissible to keep the purchased items and continue to deal with the store?

Celebrating Christmas is an innovation (bid'ah) and an imitation of disbelievers, so it is impermissible to accept or give gifts on this occasion. Scholars have explicitly stated the dislike of accepting gifts from disbelievers during their holidays.

If a store offers a gift on this occasion, it is preferable not to accept it if there is a concern that it might reinforce corrupt customs and values. However, if there is no concern about corrupting influences, or if there is a reason to accept it, such as softening their hearts, then there is no objection to accepting it. Similarly, if the intention is not to perpetuate the celebration of the birth and its festivities, but merely to align with the timing, then there is no objection to accepting it and possessing it, provided that it is permissible to own according to Islamic law.

There is no objection to continuing to deal with them within Islamic guidelines.