Safer America for Everyone Right Now Act or the SAFER Now Act This bill amends the federal criminal code, the NICS Improvement Amendments Act of 2007, and the Omnibus Crime Control and Safe Streets Act of 1968: to prohibit a licensed gun dealer from transferring a firearm to an unlicensed person prior to the completion of a background check; to make it a crime to knowingly import, sell, manufacture, transfer, or possess a semiautomatic assault weapon (SAW) or large capacity ammunition feeding device (LCAFD), subject to specified exceptions and limitations; to require a gun dealer to conduct a background check prior to the sale or transfer of a grandfathered SAW between unlicensed persons; to require law enforcement agencies to be notified when a prohibited person attempts to purchase a grandfathered SAW; to allow a state or local government to use Edward Byrne Memorial Justice Assistance Grant Program funds to compensate individuals who surrender a SAW or LCAFD under a buy-back program; to require a person who operates a gun show to meet certain requirements (e.g., minimum age and registration); to prohibit a firearm transfer between unlicensed persons at a gun show unless certain conditions are met; to make trafficking in firearms a stand-alone criminal offense; and to prohibit firearm sale or disposition to or receipt or possession by a person who has been convicted of a misdemeanor crime of stalking.