How to be an introvert and successful in the military?

Some great advice is already posted, so I'll pile on with this: Put on your leadership "hat" (get into your mode where you are focused on your people and the mission you all are supposed to do together), then ENGAGE with your folks. First one at a a time, focus on them, not yourself...learn about them...goals, hobbies, strengths and weaknesses. Smile...be nice. Imagine where that person would be best suited, and how you might "grow" them into your position. 10-15 minutes each. Repeat this until you are very comfortable. Then try it with small groups (3-4) of your folks at a time. You go out to a jet with a crew chief and a few techs...you already know one or two of them from step 1. Focus on the current job they are doing and ask questions. What makes it difficult? What would make it faster or safer or better quality? Ask what they are unhappy with, but don't let it become a bitch session. Don't make promises you can't fulfill. 10-15 minutes. Repeat until you are comfortable. You get the idea...it's up to you, and it's your responsibility to engage as a leader.

Introversion refers to where you get your "energy" from vs whether you are shy or not.

Introverts often love socializing, but it drains them, whereas extroverts get energized being part of a crowd.

You might not be giving yourself enough credit. Just because you aren't approachable doesn't mean you're not a good leader, or being looked over and ignored. That being said, the easiest thing to do is get yourself out of your comfort zone. Grab lunch with somebody, listen to what they have to say about their favorite topic. It won't be a fast process, but sometimes, the best way to do is to throw yourself right on in.