Citizens are required to apply for and pass certain criteria for driving a motor vehicle. Why not a similar license for purchasing guns and ammo?
For citizens to lawfully purchase guns and ammunition, the simplification of screening could be done by a central agency such as the Bureau of Alcohol, Tobacco, Firearms. A license issued by the federal government could be used to show the licensee has been pre-screened against a national criminal data base in much the same way that passengers are pre-screened against a no-fly list. The process could also employ a questionnaire and brief interview process.
With such a permitting process in place, gun retailers would then only need to ask for this license before the sale of a firearm or ammunition.