Yes, you can do a "length of stay" stroller rental, with a discount.
http://www.allearsnet.com/pl/strol_faq.htm
Just to be precise and to explain, it's not "length of stay," it's "multi-day." Length of stay makes it sound like you must pay for rental for each day you're on-site, and that's not the case. The first time you're in a park, you tell them how many days' rental you want to pay for in advance. They don't know how many days your length of stay is, and they don't care.
WDW rental strollers are available ONLY within the 4 theme parks. You can't arrange, reserve or pay for anything before showing up inside one of the parks, at the stroller rental place.
The stroller rental places each have 2 lines, one to pay and one to pick up a stroller. The very first time you go, you have to get in the pay line. You tell them how many days you want to rent. Singles are $10 per day, doubles are $18 per day, but with multi-day rentals you get $2 off per day, right up front. So you pay your total due (plus tax), and they will give you a strip of tickets ("rental passes") stapled to a receipt. They will stamp today's date on one of the rental passes.
You then go to the pick up line. They see you have one rental pass that has today's date on it. They then get you a stroller, and also ask you to put your name on a card (around 5" x 8"), which you will slip into a clear plastic sleeve on the back of the stroller, so you can identify it.
When you are ready to leave the park, you bring the stroller back to the pick up area. In the past, it used to be that if you return your stroller you get a Disney Dollar, but they don't do that anymore. So if you wanted to be very rude, you could just leave your stroller anywhere. I'm absolutely not advocating that, I'm just illustrating that there's no "official" way that you need to return the stroller.
You pay per calendar day, not per park or per visit. If you hop to another park later that day OR return to the same park later that day, you go right up to the pick up line and show them your rental pass stamped with today's date on it.
The next day you go to rent, go straight to the pick up line -- you'll never need the pay line again (unless you need a refund). At the pick up line, they will stamp a blank rental pass with that day's date, then get you a stroller, etc.
As someone mentioned, you can get a refund for any unused days. You just go to the pay line and give them the unused pass(es) AND your receipt.
Even though, by regular price, 2 single strollers ($10 + $10 =$20) are more expensive than one double stroller ($18), they are identically priced for multi-day rentals. Recall that the multi-day discount is $2 per stroller. As a result, 2 singles become $8 + $8 = $16, the same as the double stroller discounted price of $16. Midway through our last trip, our 6 year old and 4 year old were at each other all the time, so we got a refund on our unused double passes and made an even exchange for 2 sets of single passes.
The stroller rental refunds are available ONLY at the stroller places in the 4 parks, so if you bought a rental pass for your departure day and you are unsure whether or not you will go to a park on your departure day, you're probably better off getting a refund the day before. If you end up going to a park anyway, you've cost yourself $2 (because you'll have given up the discount), but if you don't go to a park you've saved yourself the cost of a day's rental in the first place.
