What if there is no purchaser? NO ONE paid for the gun.
The person who will be purchasing it is the buyer. Just as if they walked into a brick and mortar and picked a gun off the wall. They are the actual purchaser, just as it's spelled out in question a.
If person X fill out the form and pays for the gun, great. If person X fills out the form and person Y whips out the credit card to pay. Nope. Straw sale.
You did not answer my question. What if NO ONE paid for the firearm in question?