According to the datasheet here: http://www.onsemi.com/pub_link/Collateral/CAT4101-D.PDF
You can apply 3-25v to VCC. You only need to apply a half a volt or so more than your string of LEDs require. The best solution is to get another power supply at or close the the required voltage. If you can only get a few volts above you can use a voltage regulator to step the voltage down, but anything not used is dissipated as heat. Stepping down from 24v to 6v is going to create a lot of wasted energy/heat.
Keep in mind the power supply you choose has to be voltage regulated as this driver will only regulate current.
EDIT: Just noticed you were looking at powering the dimming boards. Your best option is probably to use another power supply as you have to power the driver board with a 5v source as well. You could get a 9v wall wart and a 7805 regulator to power all your boards that require 5v, depending on the current required of course.
I wouldn't want to reduce the voltage from 24v to 6v. I meant reduce BY 6v. From 24v to 18v. The reason I ask all of this is because I want to be freed form this half blue half white lighting with only 2 dimming channels. I want to have the freedom to do a 28royal blue, 22coolwhites and 8-10 of another color, have the 3 colors be individually dim-able and running on 1 (maybe 2) power supplies. That way there's less bulk to hide. But as far as I can see, this kind of setup would be hard to do if I'm stuck with having 6/7 led's per string for ALL strings.