They say you will have it a minimum of a week before your trip. I had this problem for our trip. We were actually leaving home 10 days before our Disney trip started to go to a wedding and visit family. I was really worried it wasn't going to get to be before we left. I had the items sent to the closest AAA office and I told them to call me so I could pick them up when they came in. That was a little under a week before we left, so about 2 weeks before our check-in date.
From my understanding, you are supposed to get the paperwork later than you would if you didnt book through AAA because Disney sends the stuff to them and then they send it on to you.