1) Anticipate and prepare: Before an emotional moments happens, you should already be ready to go. Do you know when they are coming? If not, I would recommend looking through your timeline you have ready for the wedding day. Will there be an exchange of gifts in the morning from the bride to the groom, and vise versa? Will there be a first look? Are there any surprises during the day (that always allows for emotion to arise.) I truly believe this is part of being a wedding photographer that can set themselves apart from others. It's counter-intuitive, but emotional moments are sometimes quite predictable.
2) Have the gear that can capture the emotion: A wedding is naturally an emotional day, but you can miss the moments if you don't have the proper gear to capture it well. There is a big difference between capturing an emotional moment with a wide angled lens, or one that allows for noise (i.e. if there is clutter in the background, you might not notice the tear), compared to a lens that can focus in on the tear, or the expression on their face. You're going to want either a lens that can capture that from a distance, or you need to get closer. Expressing emotions are intimate, and if you're not close enough, you might miss it.
3) You set the atmosphere: As a photographer, you are with the couple all day long. You are by their side for almost the entire wedding day, so you need to be aware of your attitude and behavior. Even though there is so much going on, you are busy, tired, hungry, trying to remember a million things, you also need to focus on how you present yourself, and yes the "vibe" you are giving off. If you are stressed out, it will make your couple feel stressed. It will mute sincere emotions. If you're calm, then that will allow for room to draw out emotion. Again, expressing emotions is an intimate thing, and if your couple is uncomfortable in front of you, they're not going to express those emotions.