Deployable badges are earned by getting kills with deployable weapons. Badges begin to unlock a short way through the single player story. Make sure you've actually unlocked a badge in the base game before attempting this challenge. Deployables include the following:
Acid Sprinkler
Taunt Bot
Proximity Mine
Pulse Mine
Turret Copter
This challenge can be completed in a Buck National vs The Apocalypse challenge, which there are four of. You can obtain this very easily (as well as Title Update DLC achievements
Title Update 2 for Sunset Overdrive) and should get all your badges within it. I would recommend the 'Buck Strikes Back' challenge specifically.
Alternatively, you can load up any night defense mission for an equally easy method of racking up kills.
Personally, I will be using Pulse mines as they stay in place for a good amount of time and you can shoot many of them at once. Simply equip your choice of deployable,
as well as the Second Amendment amp if you have it so you don't run out of ammo, and spam it at all the OD. I highly recommend using this amp or have a backup deployable weapon in case your run out of ammo.
That's it! Please vote if you found this helpful, and comment if you didn't.