No, it is not a Straw Purchase.
The laws relating to Straw Purchases of Firearms (federally) are contained in 18 USC 922.
Paragraph 6 states:
It shall be unlawful.....
for any person in connection with the acquisition or attempted acquisition of any firearm or ammunition from a licensed importer, licensed manufacturer, licensed dealer, or licensed collector, knowingly to make any false or fictitious oral or written statement or to furnish or exhibit any false, fictitious, or misrepresented identification, intended or likely to deceive such importer, manufacturer, dealer, or collector with respect to any fact material to the lawfulness of the sale or other disposition of such firearm or ammunition under the provisions of this chapter;
That's what makes it a crime on the buyers end. If you lie on a 4473, or in a spoken statement to the FFL it's a crime. But you can see that only applies to trying to get a gun FROM a FFL.
18 USC 922 (d) covers the types of person you can't knowingly transfer a firearm to as a private party sale. You can look it up, but it's the same list as the questions on a 4473. Felon, Crazy, Pot smoker, etc.
So, by federal law (your state may vary), in a Private Party sale with no FFL's involved, there is no requirement for the seller to ensure anything other than the buyer is a resident of the same state, and there is no reasonable cause to think the buyer is a prohibited person. The buyer can then take their new firearm and do whatever they want with it, including sell it to someone else, as long as that second sale is also legal.
Example:
I'm at a gun show and see a private seller with a G19 at a good price. I know my good friend is looking for a G19, and he is legal to own guns. I can buy that pistol, face to face
planing to sell it to my friend. I then take it to my friends house and sell it to him face to face. This is legal federally.*
Example 2:
I have a friend that has a felony record, but lives in a bad 'hood. I buy a gun face to face legally, planning to transfer it to my friend. I then take it to him and sell it. The first transfer (to me)is legal and the original seller is fine. The second transfer is illegal, because I know it's going to a felon.
Bottom line is that the term "Straw Purchase" as it relates to federal firearms laws, only has to to with transfers from an FFL. Private Party transfers have a different set of restrictions to follow, but they don't include that particular one.
*Caveat: If I were to do the described transfers more than once, at some not very well defined point I would be engaged in a business of selling firearms and would need a FFL. But that's a different can of worms.