Only explosives and high caliber bullets can kill Juggernauts. They appear only in Spec Ops in Snatch and Grab, Estate Takedown, High Explosive, and Armor Piercing.
The first Special Ops mission you can get this on is Snatch and Grab. You can play on Regular to make this easier. When you first start, you will have the Intervention as your secondary weapon. Switch to this weapon () and then wait for a Juggernaut to appear. There should be a one at the start of the mission but if not, you can either progress through as normal or restart it. Once you see one, shoot it five times and you should kill it. You can also use 4 RPG rockets to kill it or a full LMG clip.