To unlock this achievement, no player can pick up any gun throughout the run. Avoid any guns on pedestals, and when you defeat Granny the achievement will unlock.
This means that you can only damage enemies and bosses by dashing.
The trick is having a second controller plugged in. At the beginning of each level, players exit the worm one at a time. If player 2 remains entirely inside the worm (not peeking out), and player 1 dies, player 1 can press A to respawn again. Thus you effectively have unlimited lives to complete the game with no gun.