Can I write to a (prestigious) conference that I am interested in chairing a session?
Yes, it's okay to volunteer to chair a session, but there is not a lot of benefit to it and you might be turned down.  People may see it as unusual, since it's not a prestigious job (certainly nothing to list on your CV).

It's important to have a chair who can keep the session running on time, provide appropriate introductions, and sometimes ask a question if there are no questions forthcoming from the audience.  These are fairly easy tasks, but if I was organizing the conference and I didn't know you, I probably wouldn't agree to let you chair since I don't know how capable you are in these regards.