"Buy it" is undefined. If "Buy it" means "Purchase it with your name on the 4473 at an FFL shop and then go to another FFL to ship it to me with no 4473 at the other end", you could possibly argue it was. Since he's talking about shipping it to a local FFL, "Buy it" clearly means "Give the guy the money and then have it processed through to a local FFL where the person who wants the gun will fill out the only 4473 involved in the transaction in his own name", so, no, it clearly it is not.
EDIT: Going a bit further, a solid argument can be made against it being a straw purchase even in the first case above. From the ATF website:
"In the United States, federal law prohibits certain individuals from legally purchasing firearms. These restrictions are designed to prevent guns from falling into the hands of those who might pose a danger to themselves or others. A straw purchase is the illegal buying of a gun by an individual, a “straw buyer,” on behalf of such a person."
In the case given, as the person who wants the gun is not prohibited from buying a firearm, the purchase is not on behalf of a prohibited person and therefore fails the definition in the third sentence. Then again, it doesn't always matter what the law says, it often only matters what the drunk ATF agent at the bar thinks it is at the time and how far he's willing to go to protect his ego once he's sober.