Most printers will do that. I did a little concertina brochure a while back, a lot like that one. The printer I used printed it then folded it for me. Just ask the printers you usually use if they'll fold it for you.
Depending on the orgiinal size... lots of printers may bot be able to do it! It's often called miniature or medical print-finishing and is done by specialist trade-finishers. But most printers would have a trade-house they would use for that.