To the OP --
I am 100% certain that getting braces isn't vain, and certainly not sinful.
You might as well say that brushing your teeth is sinful, because it's "worshipping your body" -- after all, hundreds of years ago people commonly lost most of their teeth by age 40 due to tooth decay.
Sorry. It doesn't work that way.
God gave us a body (one saint called his body Brother Ass), and we can spend a reasonable amount of time taking care of that donkey. But what reasonable man would spend hours a day taking care of his donkey? Or dressing it with expensive clothing? But on the other hand, you WOULD feed it, water it, make sure it didn't get sick or die prematurely, etc.
You wouldn't pamper a beast of the field. You wouldn't prepare exquisite meals for it to thrill its palate and give it every possible taste experience. Long story short, if you consider your body like a beast of burden, that should give you some perspective on what things you SHOULD do to take care of your body, and what things are vanity, gluttony, or even idolatry.
And another point: for most people, taking care of a donkey would be a CHORE, even a BURDEN -- not something you avidly look forward to every day.
If you NEED braces, that's different than WANTING them. It should be for the sake of FUNCTION (not having tons of pockets where food settles and decays your teeth, etc. You have to be able to clean your teeth so they don't rot. Just having imperfect teeth (a single gap, for example) isn't a big deal. But a toothbrush isn't a miracle worker -- there are limits to its reach.
But when you have crowded teeth, with various teeth overlapping and forming spots where it's near-impossible to clean -- that's something you need to get fixed through pulling some teeth, and then braces to straighten the teeth out to fill the gaps. My teeth were crowded in this way, and so I can understand that there is a medical need for some people to get braces. In my case, they started by pulling 8 teeth (including 4 wisdom teeth) and you wouldn't know it to look at my smile today, because there are no gaps. That's how many extra teeth I had! Sometimes genetics throws a monkey wrench in things.
God gave us teeth for a reason. We need them to properly consume and digest our food.