There isn't an inventory system for your dorm itself. Instead:
* You place items that are specifically designed for your dorm. These are usually earned through quests, crowns (the in-game currency), or events. When you acquire such an item, it will often have an option to place it in your dorm directly. They appear in your inventory as regular items, but have a placement option that's specific to your dorm.
* You can't place gear or other regular items. Your wizard's gear, spells, potions, etc., are kept in your inventory and backpack, separate from your dorm.
In short, think of it less like "putting things in" and more like "decorating with pre-selected dorm items."