Bars go through the platform and sit on a wave table underneath like this:
|--|-|---|--|
|/\/\|/\/\/\|/\
The bars for the horses are the |s and as it spins, the bars go up and down on the waves(which are rounded). I know that this isn't how a real one works, but it would work for a miniature.
The one I used to ride on as a kid had the top of the horse pole attached to a horizontal bar that rotated. There was a 'U' bent into the bar where the horse was attached. As the bar rotated it pulled the horse up and down with the 'U'. You could use a gear system like a car differential but with more shafts to drive each horzontal bar as the merry-go-round turns. What would be the drive shaft in a car would be attached to the ground. As the m-g-r turned around the shaft it would actuate all the horses.