Kelly told CNN that it was "important for me to have firsthand knowledge about how easy it is or difficult it is to buy a weapon like that."
Kelly and Giffords have become high-profile advocates for gun control after the former lawmaker was shot in the head during a shooting rampage in Tucson in early 2011.
On Facebook, Kelly said he planned to turn the weapon over to the police.
"I don't have possession yet but I'll be turning it over to the Tucson PD when I do," Kelly wrote. "Scary to think of people buying guns like these without a background check at a gun show or the Internet. We really need to close the gun show and private seller loop hole."
Kelly and Giffords have made a strong push for lawmakers to institute heightened background checks on all firearm purchasers.
Congress and President Obama have pledged to address the nation’s epidemic of gun violence in the aftermath of December’s mass shooting at an elementary school in Connecticut. Many proposals, including calls to ban the sale of military-style assault weapons and high-capacity clips, face an uphill struggle in Congress with heated opposition from the nation’s gun lobby.
But gun control proponents believe background check legislation has the best chance of passage in this session.
However, progress on a bipartisan agreement appears to have stalled after GOP lawmakers raised objections that the proposal could lead to a federal registry of gun owners.
On Wednesday Sen. Charles Schumer (D-N.Y.), who had been crafting a background check bill with Sens. Tom Coburn (R-Okla.), Mark Kirk (R-Ill.), and Joe Manchin (D-W.Va.), said he would move ahead with his own background-check legislation after Coburn walked away from the negotiations.
Coburn said he could not support Schumer's push for gun sellers to keep records of their background checks for gun purchasers.
In his Facebook post, Kelly argued that the current background check system is far too lax.
"Even to buy an assault weapon, the background check only takes a matter of minutes," Kelly wrote.
Giffords delivered emotional testimony before a Senate Judiciary panel last month on gun violence, urging lawmakers to act. Her PAC is also running ads pressing lawmakers to enact background check legislation.
Last week, Giffords visited the Safeway supermarket in Tucson where she was shot.
Giffords said Congress needed to be "bold, be courageous, please support background checks."