I'm not aware of any either. That being said, taxis are plentiful and not that expensive to any downtown location. Additionally, there are several CTA stations within an easy walking distance that can get you almost anywhere in the city. (Be advised that the closest CTA stations do not have elevators though, so it can be a bit of a hassle if you have a lot of luggage).
I don't believe so - but I think that's because there is no need for hotel shuttle buses in downtown Chicago.
It's already too easy to get to nearly every hotel in the Loop, River North, Mag Mile or Gold Coast. The CTA 'L' trains (only a couple blocks from Union) will get you almost anywhere and provide an easy connection to anywhere in the Loop, River North - cabs are plentiful around the station - CTA buses connect all around Union - and the Chicago Water Taxi connects with trains on the north side of Union (Madison St) taking you to the Mag Mile for $3.
If you're staying at any of the hotels along or near Michigan Ave., north of Adams, the 151 Sheridan bus stops right in front of Union Station (the old building on the west side of the street).
The bus stops all along Michigan Ave and continues on to the far north side, runs 24 hrs, and it's impossible to miss Union Station as the bus originates and terminates there.
Easy to get on with reasonably sized luggage - we do it whenever we're overnight in the city.