Ordinarily, I'm a liberated confident female. Really I am. But I don't make the first move. I will make the next several, as several others have said, but I really freeze up when it comes to making the first one.
In my case it has less to do with my gender (I think) than a general twist in my character left over from crumby parenting. I have a tough time making any sort of invitation with someone I don't already know extremely well. Imposing where I'm not wanted is something that terrifies me way down deep. Even here, I sometimes get nervous about sending an unsolicited PM. So a first move is just going to get stuck in my throat. In some ways, the worst part of it is that I'm SO good at covering up my little issue that very few people realize it's there. I can flirt like anything, it's different in an important yet difficult to describe way. But even online, anyone who's played with me here knows they had to make the first move. I just can't do it.
So, keeping in mind that my own answer is an admittedly extreme and unusual case, I wonder sometimes if the typically female reluctance to make the first move might be something similar? We aren't supposed to ask for things, we're supposed to supply things for other people's needs. We're nurturers, not needers. Or at least, womanly women are. If it's true, then asking for approval or affirmation should be difficult to. It rings true to me, but I'm curious to hear if other's agree.
Having confessed, though, I'm the first person to put my hand up and say it's not fair. Guys have egoes that can be bruised by rejection. They fear risk as much as we do. I can't imagine it's EASY for them to make the first move, even if it society does dictate it be their job. But since I can't share the burden, I make sure I'm kind and appreciative of anyone who does take the trouble to let me know they're interested. And that I'm gentle if I can't reciprocate the feelings. I'm well aware life'd be much more boring if they didn't do the heavy lifting at the outset

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G