You will have to call each department and ask. I was in the same situation last week (luckily a little more pushing and I got them all in). Anyways, I called my departments and it varied for each one, and I had 3 due on Jan 1. One school told me it would just need to make it by mid-Feb (!), Another told me if it got there by Jan 9 it would be OK, and the other said it wouldn't really matter, but my app wouldn't get reviewed until it arrived, so it needed to be there by the time they made the last round of acceptances, whenever that happens to be.
They were all understanding and said they deal with it a lot, but I would definitely recommend calling so they are aware, and so you can relax a little because they will most likely tell you there is a grace period for it as long as the rest of your app is in. I also included a cover letter in all of my apps listing what was included in the packet, and made sure to note that a third letter had been requested and would be arriving (or uploaded) separately.