Yes, Disney does allow the motorized carts on board the ship; however, just like in the parks, people are EXTREMELY rude and constantly "cut in front" so it is very, very stressful trying to get anywhere... as long as you take your time, and breathe deeply to control the desire to "floor it" and deliberately crash into the person

, you'll be fine. Also, the elevators are incredibly small and, again, rude people make the experience unpleasant by basically laying/pushing on the cart so they have room.. hello... truly unbelievable experience the first time--after that, you sadly get used to people with absolutely no manners.
I would suggest booking a handicapped room to allow room for storing the cart as we had a "regular" cabin one time that required my mom to park the cart in the hall. Noting "rude people" once again, we were constantly surprised how many people let their children climb on the cart as their own personal playground... sometimes to the point we'd open the cabin door, stare at them for literally about a minute before they finally wised up.. and almost no one would ever apologize---they would actually look at us with frowns like we were intruding on their personal family "fun" experience... some people are truly clueless!
You may also want to consider renting a cart from a company in Florida. We've found over the years that it's easier to have a company deliver a cart to the hotel (we do land/sea packages) and they'll pick it up at the port at the end of the trip---although it adds a few hundred to the price of the trip, it's easier than dealing with the airport as the cart has to be broken down into pieces for easier transport by the airline, and gets really roughed up on the flights.
Also, if you plan to rent in the parks, make sure you get there EARLY. Magic Kingdom only has a certain number of carts, and often runs out (found this out the hard way twice!)... and if you leave during the day, there's no guarantee one will be available when you return at night... and if the cart is left unattended, they will "confiscate it".... these fun experiences led to rental for the entire trip from private companies...
Not to sound totally negative---Disney does do very well at the bus system---they are always quick "uploads" with the carts, and I'm grateful they offer the service!!
All in all, carts make it possible for a great experience for handicapped people... but it would be great if public service announcements were made to the general population to be kind to the physically challenged and be respectful of people on carts!