It all depends. In theory, you should only get there within an hour of your crossing - they don't want the parking area too busy. We have never been early going out, but have been a couple of hours early for our return on two or three occasions. Each time, the person in the check-in booth has let us through with the sticker for the next available shuttle - very helpful. Don't know whether the automatic booths would be as helpful, though!!
You could try giving them a call, or just turn up early with a prepared "very sorry, but..." story, smile and say "thank you".
If you are not crossing at a peak time, there will almost certainly be space on a shuttle.