Disney does not have any events geared to Mardi Gras but you will probably see Saints or LSU jerseys or purple, green and gold. It does get crowded because so many from LA, MS and AL head to Disney. In NOLA area kids are out iof school all week.
FYI - Mardi Gras in New Orleans is not all the craziness you may see coming from the French Quarter. Those with families are out enjoying the parades along St Charles Ave Orin the suburbs. It is mostly tourist in the Quarter.