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.