This trick works once you have Dogmeat accompanying you (which you can also befriend shortly after leaving the vault). This weapon is inside a locked weapon crate which can only be opened conveniently if you have a high lockpicking perk level. However, you can still get this early as now.
Return to Vault 111 (the controls to the lift is found in the small chamber nearby) and head to the Overseer's office. This is easier done when you have looted all possible items in the room. Guide Dogmeat near the security case where the Cryolator is located then order him to fetch you the item. It may take a few attempts but he will get it for you. After successfully getting the weapon, talk to him again and Trade to get the Cryolator's ammo. This is a very powerful gun early on and its ammo is quite a precious commodity so save it against harder targets.