We have a different definition of "Sick bucket." DW and I are the infirmary for our DGDs (now 5 & 6 yrs. old.) Whenever they stay home from school/daycare they come here to recuperate. We have a tupperware type bin filled with little Dollar store trinkets, coloring books, puzzles,etc which is known as the sick bucket. If ya gotta be sick at least you get to choose an item from the "sick bucket" to divert the misery for a while.
With regards to the original question, usually some kind of plastic or mixing bowl. Haven't needed it for quite a while. At some point they get old enough to run to the bathroom.