They do because I used them to move to the UK and I use them to mail things back to the States. They don't have too many things on a homepage but they do mail things worldwide.
They do have a form on the page where you tell them what you want and they mail you back a quotation. If you are in London, I suggest you go to the Mitsukoshi branch because they will walk you through it step by step for those who are package-challenged like me. Also if you need boxes, they will give you boxes and packing bubble thingies to take home.
If you want to know for sure, I would call them up and ask.
I just checked their ad in this Thursday's JP newspaper and the rates of what I wrote in the post above still applies. I think if you are only goign to mail one box, i woudl not bother with Yamato but if you are mailing more than 5 boxes, I would definitely consider them.
I will note though if you live in a State where there isn't a branch , it might cost you more because they will have to drive out further to get to you. And since I am really L......acking motivation

they even unpacked for me. No I'm not kidding, they unpacked for me too. (Do I win the award for ultimate L...acking motivation?)
I will note if you are doing it around the holidays, it will take more time for things to get there with everyone else sending packages too.