I would start with the packing list that Disney has on the website, the one that you can log into where you accepted the offer. That is a great place to start. But for a quick list off the top of my head.
1) Twin Size Sheet
2) Bathroom Linens
3) Toiletries
4) Lock (Pad or combo for the lockers if you have things you want to lock up)
5) Alarm Clock (Dont want to be late for work)
6) Money (Dont get paid till 2nd week)
7) Drivers License (Guess u will have that seeing u are driving)
8) Documentation for USA employment (There is a list that says (1) thing from list A or (1) thing from B and C) (i.e. Drivers License and SS Card is (1) from B and (1) from C)
9) The paperwork in "The Purple Folder" (incase something happens)
10) Casual / Dress Cloths
That is 10 things I can think of. I would say anything that you seem to have or need on a day to day bases at home, you may want to bring that with you. The website they provide that you logged into the accept has alot of information of what to being, what to do, and what not before you go. Thats a great place to start.
Storage under bed Question. If I remember right, they do have access to under the bed. They are not the beds like in college dorms that are on the posts where u can raise them high to get alot of extra space under, they are just standard twin beds that have about 1ft clearance.