There are essentially two methods to get the Mandibular Rearranger in The Outer Worlds. You can either follow a quest chain that will lead you to it, or you can just travel to the location where it is and pick it up.
The quest that leads to the Mandibular Rearranger and the other Science weapons in the game is called "Weapons From the Void", and you can get this quest by purchasing an item called Hephaestus Mining – Archive Cartridge from Gladys on The Groundbreaker (she is the same NPC that sells the Navkey to land on Stellar Bay on Monarch). The item costs 300 Bits without any discounts, but you will probably get it a bit cheaper.
When you have the item, head back to the Unreliable and into your captain's quarter and use the computer there. There are now some new options on the computer where you can research the mining facilities on Scylla. When you have read this, the quest will update and give you the marker for the Mandibular Rearranger.
Now travel to Scylla and head towards the quest marker in the area called Abandoned Mining Camp. While it's on the other side of the map, Scylla is not that big, so it only takes between 5 and 10 minutes to get there, depending on how long it takes you to beat all the enemies on the way. The Mandibular Rearranger itself is just inside a house, so just enter it and pick up the weapon, and it is yours to keep forever.