Strollers are $8 per day to rent. They are jogging style strollers, and there are NO doubles- all singles. Be sure you save your reciept and the blue card it is stapled to. If someone takes your stroller, you leave for a while or switch parks they will replace your stroller if you show them your receipt. As the others have said, your name will be on your stroller.
When we were there last Oct you could not take the strollers out of one park and into the other. You had to get a new one when at each park. When we went earlier this month, you could take your stroller out of one and cross over to the other. Much easier.
Stroller can be rented near the main enterance to DL or at the star trader store in tomorrow land. You can't see them in the star trader store- but you can rent them there. They store them in the back room. You'll want to go to the cash register in the back, near the exit from star tours. In DCA you rent them near the enterance- next to the locker rentals and bathrooms.
Do NOT lock your stroller to a fence or anything else. Security will cut the lock if your stroller needs to be moved. If you feel like you need to lock it, use a bike lock and lock it to itself. Run the bike lock thru the wheel and around the frame. CM's will still be able to move it if needed, but its going to be hard for anyone to steel. But in reality, its really not needed. Very few strollers are ever taken at DL.