Yes they allow scooters. No, they won't rent them to to you. There are several companies that will have one delivered to the terminal area for you.
If your father can walk in the cabin, you may choose to have a "regular" cabin rather than handicapped. The accessible cabins are larger but are missing some features that are present in the "regular" ones...and having just done a cruise with a friend in a wheelchair, I'd like to avoid an accessible bathroom for a while! If you choose a "regular" cabin, the scooter won't fit into the cabin well, but your stateroom host will take it to one of the "cut out" areas each night and recharge it for you.
If you request an accessible cabin, the scooter will fit in the cabin...and you'll have to take care of the recharging. Many people who don't actually need the handicapped facilities find it easier to use a scooter or wheelchair on the ship because of the distances involved. If he needs the accessible cabin, have him complete the medical form and get his doc to sign it...then fax it to the number on the form. If he doesn't need the accessible facilities, you don't have to do anything.
A third way to go is to note that he will have a scooter but doesn't require the accessible room. He'd like it if it is still available as the sailing date approaches but doesn't want to take it from someone who truly requires it in order to sail. If he chooses this option,
DCL will note it on your reservation and move him about 30 - 45 days before sailing.
A final thought if a wheelchair would do instead of a scooter...my friend has a passion for having chairs available to people who need them. She routinely purchases them at Goodwill for $5 - $15 and donates them to large churches and other facilities that she feels should have them available. At that price, you could buy one to use on the cruise and get rid of it afterwards if you have no further use for it. It's cheaper than renting!