Hi Christine ~
There are double
stroller rentals available in all of the theme parks. Downtown Disney only offers single stroller rentals.
Single Stroller Rentals are $15 per day or a Multi-Day (Length of Stay) Rental is $13 per day.
Double Stroller Rentals are $31 per day or a Multi-Day (Length of Stay) Rental is $27 per day.
The link above will give you more details on rentals but keep in mind that prices are always subject to change without notice.
The strollers are made of hard plastic and do not recline so napping would be difficult. I would bring a towel or small cushion for your kids to sit on. The small kneeling cushions you use when gardening work great. They are small enough to put in a backpack and they dry quickly if they get wet.
The strollers do have a canopy and have a little (not a lot) of storage space. The picture that is shown on the stroller rental link on the Disney website is not a picture of the strollers for rent in the parks. You can see a picture of the single strollers and double strollers on the website AllEars.net. If you search for "strollers" on their homepage, the first link shows a picture of both.
Another option is renting from a private company in Orlando. There are some local companies listed at the bottom of the Stroller Rental link on the Disney Website. They will deliver to your resort so your stroller is there when you arrive and pick it up at the end of your trip. A local rental company may even save you money versus renting in the parks.
I answered a question for another Guest with some
tips on strollers that you may also find helpful.
Wishing you minnie magical memories :-) Allison V.