I've had some quiet players from time to time over the years, often in pickup groups. (My regulars are very comfortable with each other and have no issues with making themselves heard.) I do view players who are too quiet as a problem in a game like D&D and I try to address it before play.
First, I explain that a game of D&D is pretty much just a structured conversation between the players and DM. I say a thing, you say a thing, I say a thing, repeat. Sometimes there are dice involved to help me decide what to say. In a conversation that isn't about elves and dragons, if one person is doing all the talking, it tends not to be a very interesting conversation. So to make it an interesting conversation, we need everyone to engage in it, each making about the same contribution during the course of the session. I try to spotlight people about equally to draw people out.
Next, I make sure that the players are aware of how to talk to each other in a way that is productive and doesn't shut other people down. I'm sure we've all had people in our lives that are eager to tell us all the ways something won't work or starts nearly every sentence with "Well, actually..." or just love to poke holes in things. Flumph those people. Someone who his already shy is likely to shut down when faced with this kind of person. Why offer ideas at all if other people are just going to shut them down? Better to just stay quiet and go along, right?
So, I tell the group that when they are hearing an idea or proposal from another player, they should try to find the good in it and think of ways it can work rather than ways it can't. This may mean adding to the idea to make it better. They shouldn't shoot down the ideas of others outright - this just creates debates that stymie play or causes other players to withdraw from the conversation. Instead, they should work together to take a decent idea and make it better by adding to it rather than taking away. When I see a player start to criticize, I jump in and say how I (the DM) think the idea could work and encourage the criticizing player to suggest ways to improve upon the basic idea to increase their odds of success in a manner that doesn't negate their fellow player's idea.
As well, I'm always quick to give praise and reward for good ideas. A simple "That's awesome," a high-five, or some Inspiration is great positive reinforcement for a shy person. And in my experience once I start doing that, other people do too, which makes for a very positive, supportive atmosphere at the table. It's good to see the players cheering each other on.
These approaches, taken together, are very simple to implement and I've seen introverted players jump right into things. When they come to the realization that their ideas are going to be supported rather than criticized and that everyone else wants to hear what they have to say, it becomes a lot easier to come out of their shell. And even if you don't have shy players, these approaches certainly make the game experience that much smoother.
