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.
I've never heard of the term "straw buyer", and Onelook lists only one dictionary that contains the term. That dictionary is Wiktionary, which can be edited by anyone and is somewhat notorious for inaccuracies and a generally very permissive attitude toward including words. (Wiktionary's entry does not include any sources, references, or examples.)
However, see the Wikipedia entry on "straw purchase"; that might suit your needs.
Another word that you could use would be "agent". From M-W:
4: one who is authorized to act for or in the place of another
This does not always involve making purchases, but it certainly can. There are also related terms, such as "purchasing agent" and "buying agent", although they can have different meanings in some industries.
You might also be able to use a related term such as "broker", "intermediary", or "middleman", but those folks often simply arrange transactions without purchasing items themselves.
Finally, I'll note that terminology can vary greatly across different dialects, in different legal systems, within different industries, etc.
The term would probably be a proxy buyer. A 'proxy' is someone appointed or authorized to act for another.
'Straw buying' is something more specific - where a person cannot buy for themselves, perhaps for legal or financial reasons, so somebody else purchases it and then transfers it to them. It might be to avoid taxation (tax 'avoidance', as opposed to 'evasion' can in some cases be legal).
Certain specific areas of business may have their own terminology for such a person - for example, the terms 'buying agents' or 'purchasing agents' are used in real estate. Other terms, such as 'broker', sound much more official and one would expect these to be making money out of the transaction in the form of commission.
