For this reason, I did a lot of communicating via email. I felt like I wasn't doing as well as the more extroverted types who were more obviously able to suck up to our immediate superiors.
Then me and some of my teammates were moved to a new team. My very professional and established track record of sending emails most of the time meant my questions were still getting the answer I needed. The seemingly successful extroverts were incredibly frustrated and tearing their hair out. They didn't know how to function in the new conditions. They began turning to me for advice on how to get things done on the new team, physically separated from the people they had established relationships with.
Learn to play to your strengths. Being an extrovert isn't necessarily all it is cracked up to be. Most of those so-called friendships are more shallow than you think.
If you can find a nice and sympathetic extrovert, they may be able to do some introductions and the like for you and bring down your stress levels. I did that sort of thing sometimes for a painfully shy teammate. But, beyond that, don't believe too much of the hype.