When X buys something, they are a buyer.

When X sells something, they are a seller.

What would they be called if they buy something on someone else's behalf?

A straw buyer???

Note that X does not intend fraud that involves in straw buying.

X buys something for Y and knows that Y will like it and pay X the amount X has given from their pocket, and then that thing bought by X will be handed over to Y by X.