The Shulhan Aruch in Siman 450 permits giving money to a non-Jew on Pesah and instructing him to go buy himself something to eat, even if he is likely to buy Hames. However, he may not tell him explicitly to go buy Hames. Also, one may not tell the non-Jew to buy Hames on credit from the store and offer to pay for it later on his behalf, since that may be tantamount to the Jew owning the Hames.
For example, if a person has a housekeeper working in his house on Pesah while he is away, it is preferable that he give her money to buy her meals before the holiday. He can also tell her to buy food and he will pay her back directly, but he may not tell her explicitly to buy Hames or that he will pay the store back.
Conversely, it is prohibited for Jew to buy Hames for a non-Jew with the non-Jew’s money. For example, if a Jew was standing online at the cashier and the non-Jew gives him money asking him to buy him a sandwich. This is prohibited for two reasons. First, there is a concern the Jew may eat some of the Hames before he passes it on to the non-Jew. For that reason, the Mishna Berura forbids even touching Hames on Pesah. Second, a non-Jew is not Halchically capable of appointing a Jew to be his Shaliach (agent). Therefore, when he buys the Hames, he is the one taking possession.
