I agree with Jill on this. We're currently talking to various internet marketing experts for a book that's due to be published by London based publisher Kogan Page towards the end of this year. So far people have been pretty enthusiastic to get involved / contribute material in return for appropriate attribution / acknowledgement in the book. Having a publishing deal for your book in place when you approach them will help.
As for how you should approach them, drop them a line by e-mail or give them a call. The thing about well known experts, or thought leaders, is that the very fact that they are "thought leaders" means they're not backwards in coming forwards with their ideas and opinions. You don't get to be a thought leader by being shy and retiring (expert maybe, but well known? Not so much). Most are easy to contact via various communications channels, and many will be happy for the opportunity to raise their profile some more.
Remember you're not begging for contributions, you're entering into a mutually beneficial arrangement. You scratch my back,I'll scratch yours, sort of thing. You get material for your book... the experts get broader exposure for their expertise... and, if you pick the right experts the readers get a more compelling end product. Everybody wins.
Good luck with the book, and hope this helps.