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
2009 - Enchantment of the Seas
2010 - Freedom of the Seas