Most cruise lines (including RCI) will do laundry for a fee. You do not have access to their laundry facilities, nor are irons permitted in the staterooms because of a fire hazard.
On my last cruise on the Enchantment, there was one night (I forget which night it was) where they would do all the laundry you could fit in the bag for one flat rate, as opposed to charging by the item.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)