The shooter was only 20 and couldn't own guns himself. They were registered to his mother, reportedly a school teacher. Since most women don't own "M4 type assault rifles" (which Lanza took, but left in the trunk of his car) or Glock/Sig Sauer semi-automatic 9mms, it won't surprise me to find out that she bought them for her son to use, but that is speculation.
The point is that I'm certain his mother would have passed any background check you could put her through. In fact, Connecticut has some of the toughest gun control laws in the U..S.
Most of these American mass murderers (James Holmes, Adam Lanza, the Va Tech shooter, etc.) are young males of high school or college age. They're usually described as "socially awkward" or perhaps as "loners".
There were plenty of socially awkward loners around when I was growing up and plenty of guns too, but you didn't have this problem.
There's something wrong in our culture, whether it be violent movies or violent video games, or whatever, that causes these boys to decide that since their lives aren't good, they're going to go out in a blaze of gunfire and take as many people as possible with them.