Yes, there are companies who will rent and deliver to the port.
Walker Mobility is in the Orlando area and quite popular for rentals at WDW, and they will also deliver to Port Canaveral; this may be a great option if you are doing a land/sea combo. Also
Brevard Medical has delivered to Port Canaveral cruises for years. And
Special Needs at Sea delivers to numerous ports including Port Canaveral. To my knowledge, the
ECV will be waiting for you in your stateroom, unless that recently changed.
As mentioned, stateroom hallways can be narrow. In addition to housekeeping carts, people leave room service trays outside and sometimes people park strollers,
ECVs or wheelchairs even though they are not supposed to do so; all this creates very tight space to get past. If you do not have an accessible stateroom booked, you probably won't be able to fit the ECV through the doorway to your stateroom, and even if you do there is nowhere to go with it due to room layout;
DCL asks that all ECVs be parked in designated public areas if not in your stateroom (meaning not in the hallway outside your room) and that location will depend on which ship. You may need to have someone else in your party "valet" for you as it could be far from your room or even on a different floor.
There is a decent lip at the external doors to decks. Main hallways in public areas should be plenty wide. I recommend trying to stick to forward or aft elevators when you can as those aren't quite as busy. But they do get crowded anyway. Consider riding in the opposite direction and then back again if there is space in the elevator -- for example, from the main atrium ride down and then back up again to the pool deck. Within MDRs and lounge spaces can be tight as well. Are you able to walk at all, like to park outside the MDR and walk in to your table?
The first night I recommend that you arrive early at the Walt Disney Theater to find out the process for parking your ECV and accessing seats. If you need to ride the ECV into the theater to your seat, I believe they will take you in a side door a few minutes early and then park the ECV for you.
Castaway Cay is not a tender port so it shouldn't be any problem getting on/off the ship. The only issue at most ports may be the steep incline of the ramp, which I believe can change based on water level and which deck is used for debarkation that day. Grand Cayman is always a tender port; I'm not sure which itinerary you are cruising. At CC, be aware that an ECV isn't going to work in the sand so you'll have to stick to the paved pathways. DCL does offer beach wheelchairs on a first-come/first-served basis.
Enjoy your cruise!