That should get you through the first week, but not much longer if they're able to get the cardboard wet. You definitely want to use pine shavings for that for the absorbency they have.
I couldn't find a video that shows what they do with the water in the drinker when they really get into it. They're not that bad the first week, like these. But note the wire bottom below the waterer.
When they get older, they start zooming through that water "sifting" in it, throwing it pretty far. Chickens kick bedding into the water and make it gross. Ducklings throw the water all over and mix their food in it, then food filled water gets flung around.
They'll run to the food, get a bill full, run to the water, sift, sift, sift, run back to the food... and so on and so forth until they're drained the water container and it's contents are all over the bedding and walls. So the cardboard isn't going to last long unless you place the waterer in the middle, well away from it.