It turns out that Yeti coolers are very buoyant and do indeed float quite well even when they are completely full. This is because the plastic and insulation they are made with is much less dense than water and has lots of air pockets in it.
The answer is YES and they are actually quite buoyant. To test this I completely filled my Yeti 45 with water and put it in the bath. The water completely filled the inside of the Yeti more than anyone would naturally fill the cooler with ice or food.