I am going to talk about these points to help you understand the approach. After that, its all YOU.
So what exactly you need to do to climb the stairway to heaven ?
1) Never ever depend on anyone to carry your conversation for you : Believe me when I say this, this is your fight, you can't be dependent on anyone.
Your thoughts : "Easy to say, how the heck can I implement that"
Answer : RESEARCH
Confused ? Let me explain
When you know your audience and the company, you will never need a companion to give you a hand. By doing your research, you will feel at ease. This works.
2) Alumni's don't bite !
Think about it, an alumni is a person you can trust. He has a strong history with the same college that you are currently studying in. So there is a connection between both of you right ? Don't you see he will always be there to help you no matter how stupid your question is (or you think the questions are stupid). So next time there is an information session, don't hesitate just ASK.
3)Stop worrying about what people are thinking about you
One of the worst problems for an introvert is that they are always thinking too much. One piece of advice, be genuine in your approach. Ask questions not for the sake of asking questions, but because you want to gain knowledge. This way most probably your connection will complement you for your question. He may use words like "That is an interesting question" or "I am glad you asked", when this happens stop thinking about the social awkwardness and keep asking him questions, because he has given you the proof you needed. Thus his subtext says that he is not, at the back of his mind thinking "I need to find a way to get away from him and his questions"
Enough of philosophical talks. Lets get down to Business, lets talk about the actual approach :
1) For Introverts one of the best possible ways to network:
Be it family members, alumni or anyone influential. Catch Them when They are Alone
The shyness and awkwardness factor would never come into play
2) When you know someone within your family who is working in a firm you desire, before blindly reaching out to them, try to know them by asking questions about them , this is where small talk comes into play (YES YES I know that is the difficult part). Suppose you visit their house, for example flash a big smile when they greet you, compliment their apartment (everyone loves a little praise) , ask them how they started working and how did they first felt when they started working. Gradually, then start taking about yourself, say "Oh that is a great company, in fact I myself am looking to land an internship there, but I am not sure if I would be able to compete with the rest of the pool". Chances are the connection would help you, without you having to say a word. Why ? Because you showed genuine interest in their lives. Its time they show their side of interest.
3) Its more of a technique than advice. In a week try to network with atleast 50 people (yes! , you heard me right 50). And on a piece of sheet or any note rate the people you tried to network with by giving them stars. So if you networked with an HR of a firm, give yourself 4 stars.
This way you will feel good about you. Making you feel at ease as the process reaches maturity.
I will try to update this article as soon new thoughts formulate in my head.
In the mean time would love to hear your feedback.