Your portion is due on Jan 1st. Many other docs (scores included) trickle in afterward – and it’s totally normal. The scores are the easiest items too – they’re a simple data dump and the data is linked to your application automatically or at worst with a few keyboard/mouse clicks on their part.
This site has been around for about ten years. I don’t know of a single instance where someone reported that they got an "incomplete " app because scores, transcripts, recommendations, school reports were sent into the colleges in mid January. But I guarantee you that every late December and late October (for EA and ED apps), there will be thousands of posts and threads like this.
Colleges spend millions of dollars collectively, trying to get you to submit. They aren’t going to install a pit right at the finish line. At midnight January 1st when many of your peers are losing it, the admissions people are going to be sleeping calmly in their beds. There’s not some massive party at the office with people devilishly ready to hit the auto reject or delete button once the midnight hour passes.
And do you know what happens when something IS missing? They CALL you. Logical, no? Do you know why? They WANT to admit you (and your 4-5 years of tuition and fees, of course). As much as you want to be admitted to dream college X, the colleges want you to find a way to enroll you as well. Don’t forget that.
Please breathe, it’ll be all right. Good luck to everyone.