Stroller rentals can ONLY be done at the theme parks. There's absolutely nothing you can do in advance anyplace in WDW.
The above poster mentioned a set-up OUTSIDE the MK turnstiles. That is what they call their "express" rental, and as far as I know it's only at MK, and it's only for single-day rentals. Inside MK and the other 3 parks, you don't need to pay cash only. I've used my room charge each time I paid for length-of-stay rentals.
Not that you asked, but here's some other info ....
Stroller rental places have 2 areas, payment and pick-up. The first time you rent, get in the payment line. Tell them how many days you want to rent for. Don't tell them "length of stay" because they don't know how long you're staying, and they don't care. You can be staying at a resort for 8 days and choose only to rent for 3 days, for example. So just tell them how many days' rental you want.
At present, a double stroller costs $18 per day, and a single is $10. For multi-day rentals, you automatically get a $2 per day discount, meaning the true cost is $16 double and $8 single.
After you pay, you'll be given a strip of rental passes stapled to your receipt. The person you pay will stamp one of the passes with that day's date. You then take this over to the pick-up area, and you'll be given a stroller -- even when it's crowded, there's usually no line at all for pick-ups. At pick-up, you'll also be asked to take a 5 x 8 card (or be handed one) and write your name on it, so you can differentiate your rental stroller from all the others. I've always let the kids decide on a character name and I write that down. It's nothing official, it's just for your own use.
When you leave a park, you take the stroller back to the pick-up area. It won't be checked-in or anything. In fact, if you wanted to be rude, you can just take the stroller to the park exit (but not through the exit) and leave it there, but as I said, that would be rude -- I just put this out as an example to show that there is no paperwork or "official procedure" for returning the stroller.
If you are going to another park or returning to that same park that same day, you don't need to pay for another rental. You just go to the pick-up area and get a new stroller by showing your rental pass stamped with that day's date -- the CM will glance to see the date, and you're off.
If you end up not using all of the days you paid for, you can get an immediate refund for any unused rental passes at any of the 4 rental places (one in each park) as long as you have your original receipt.
Our 6 year old and 4 year old were bickering a lot about personal space our first couple days. We decided to get a refund for the remaining double stroller days and rent two single strollers per day for our remaining days. It ended up being an even exchange with the multi-day discount.
