Felons can't get guns. Period.
They can't legally possess guns.
They absolutely can get guns though--easily. In my state, people can sell each other guns on Craigslist, in their garages, or anywhere else. They are not required to do any background check whatsoever. It's up to the felon to not buy the gun, not the other way around.
FFLs require a background check, but they are not the only way guns are legally sold.
Here, no license is required to purchase a gun, and business owners cannot even stop you from packing heat unless they go through extra hoops to accommodate you by providing a gun locker and posting special signs.