I know that billionaire entrepreneur in Las Vegas has a "space hotel"
designed and built, but he is leaving it to others to build the
shuttle that will get the paying customers up to it.

Not entirely.

He is putting up some money for a prize which would be awarded to
someone who builds a suitable vehicle. A few teams have said they are
interested in winning it. The best source I could find was:
http://en.wikipedia.org/wiki/America%27s_Space_Prize

Also see:
http://www.bigelowaerospace.com/prize.html
http://www.space.com/spacenews/busin...ow_041108.html

Getting to orbit is a lot harder than a suborbital vehicle like
SpaceShipOne.