I've been NC with most of my family at one point or another over the years.
This sounds like either you need to really work on how to handle relationships or to walk away yourself.
NC isn't a game. It hurts people, badly.
If you really mean that you want to be in contact with her, then it'd be best to learn a lot about how to express yourself constructively, handle conflict, set your own limits for a relationship and decide what you'll accept from others and stick to it. Taking quick umbrage isn't part of it; walking away from a conflict situation, calming down and then trying to resolve it is.
If that point comes then you know your sister best, and can probably guess whether a letter, a text or an in person visit is best. If you're unsure, a hand written letter might be best starting off with an apology and acknowledging that you handled the situation badly and you're sorry. A genuine apology can go a long way to building a bridge; without it, things never quite heal. After that acknowledging that you have been trying to do things differently will let her know that you won't handle a difficult situation in the same way again, which might give her the confidence to have a bit of faith in you.
Are your parents healthy people to be around? You said that you'd been NC with most of your family at one time or another. It seems likely that perhaps your family run on high-conflict and high-drama and this is how you've learned to relate to one another. If this is the case (again, this is just a guess) then maybe it's better if your sister does keep her distance. If it isn't and your parents are calm folk, then saying that you see how she's staying away is hurting them, and acknowledge that it's your fault but that you'd like to repair things for your own sake, but also for theirs.
If your sister is fundamentally reasonable then she might mull this over and get back in touch, and things might be repaired. If the original quarrel hurt too much or was the final straw, or if she runs on emotion and grudges, then she might not.
If she does then you'd better be damn sure that if you get angry with her again, you handle it differently. She almost certainly won't give you a third chance.
All you can do is lay your cards on the table, apologise, and hope.