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.