We just ran the Katy this October and camped at our usual spots:

Katfish Katy's along the Missouri River (though it's up for sale again and the small camp store was not open)
Rivers Edge Campground (great sandwiches at the bar next door)
Klondike Park (very nice state run facility)

I don't care for the close quarters of the RV/tent campground a few miles east of Katfish Katy's but it's the only place one can get some home cooked Thai food along the Missouri River

The Turner Katy Trail shelter shown above is a decent bunkhouse and they were just getting ready to open up a new small restaurant right next door when we rolled thru so that should be convenient because otherwise you're on your own, no stores or cafes near by.

You can also camp at Marthasville near or on the ball diamonds and there's a terrific restaurant, Twin Gables, just a stone's throw south of that area.

Stealth camping is abundant most anywhere along the trail but it is illegal, that being said, if you choose this option, Leave No Trace so as not to alienate the locals.