I could imagine the character I'm proposing joining the group, at least initially, out of convenience. She's set ground rules, of course, and would probably be rather difficult to work with at first, but the benefits of grouping tend to outweigh the downside of working with others in MMO's. As time passes she'd probably start to stick with them out of habit if nothing else, and eventually grow attached to them, but she still wouldn't be the lovey-dovey sort except under very private circumstance.
Thinking on it a bit more, for background purposes I was considering her being rather sickly in real life, hence her fixation on video games. She puts so much effort into them because she can't do so many things others take for granted, and so it bothers her immensely when people say things like 'It's just a game.' or 'You should go out more often' or 'Get a life'. A part of her is actually glad that she can pass for normal in SAO.
Also, she isn't exactly comfortable with her sexuality yet. She knows she's gay, but actually telling the people in her life that, or even outing herself in a relatively anonymous setting like SAO is a bit much for her. (I'm not sure if it was possible in the original SAO, but if you allow it I would even have her go as far as making a Male avatar before Kayaba's whole 'you now look like yourself' trick. She chose it so she could have a shot at maybe romancing a girl without outing herself. I wouldn't actually spend any time in the Rp with her pretending to be male though.)