How's a Sim in the dorms supposed to access Household Inventory?

Your Sim doesn't have a household inventory while living in the dorms because they don't own a house so they don't have a 'household.'
This is why storage chests came with the pack. All they can take with them is whatever fits in storage chests they then put in their personal inventory and anything else that fits in their personal inventory. That's it!

However, whatever your Sim got in the mail will most likely actually show up in household inventory when they move out of the dorm and into a real house/apartment. When my Sim graduated with honors, I thought she didn't get her wall plaque/ribbon thingie for it. I was disappointed she didn't get one and thought it was a glitch. But she actually did get one! It showed up in household inventory once she moved out of the dorms and into a house. Perhaps that's what will happen for your Sim too? Just a thought. :)