Not to a prohibited person.
I think Fast and Furious was a botched attempted means of advancing a political end through illegal acts by law enforcement.
Certainly it can be sold to a prohibited person.
What would possibly stop such a sale?
If a law exists, but is unenforceable, than it is nothing but words on paper.
And whatever you think of the motivations for Fast and Furious, ("Motivations" which would be completely un-provable, in any case) requiring background checks would have negated the reason for the whole operation.
AND would have made absolutely sure that the rather unfortunate result (sales of weapons to Mexican Cartels) would never happen again.
So, to express outrage over F&F and then not allow this bill to pass is pretty much the ultimate in utter hypocrisy.
How can you express outrage that the government allowed straw purchases to criminals, and then block legislation that would stop straw purchases to criminals from happening???