Will the military provide active duty members with orthodontic braces?

In certain case they will, but as far as I know, not for asthetic reasons. If your malocclusion could possible cause you severe dental problems in the future or you are in an MOS where your ability to speak clearly and distinctly (you have a speech impediment that could be cured by braces), then the military will provide them. However, they usually do not do this unless you are going to be at one duty station for a an extended period of time (few years). If you do not fall into either of those categories, you have to obtain braces yourself at your expense.