So, you cannot even pay in advance for park stroller rental at your resort?
That is correct. In the parks, by the parks, for the parks.
But like I said, you can do it once, and then never have to wait in the "pay" line again. The pay line can get wickedly long at times, but I've never seen a stroller pick-up line be even a line at all.
The first time you're in any of the 4 theme parks, go to the stroller place and get in the pay line. Tell them how many days' rental you want (don't tell them "length of stay" because they don't know how long you're staying for, and they have no reason to know or care). They'll give you a strip of rental passes, connected by perforations, stapled to your receipt. They will stamp one of the passes with that day's date. You then take that to the pick-up line and show them. Those CMs will give you a stroller (as well as a name card to fill out, to help you identify your stroller) and you're off!
On that same day, if you leave the park and return later, or if you hop to another park, you don't need to get in the stroller pay line. Just walk right to the pick-up line and flash them your rental pass with that day's date stamped on it, and BAM, you have another stroller.
The next day you're in a park, hand them your mass of rental passes/receipt, and they will stamp that day's date on an unused rental pass, then give you your stroller (and again give you a card to write your name on). No muss, no fuss, you're on your way.
I've seen people treat those name cards with veneration, like they are deadly serious things and are worth lots of money or something. Each time we got a stroller, I'd ask one of the kids to name their favorite character and I'd write that on the card. It's just a means of differentiating your stroller from the many other identical rental strollers.
If you decide you won't be using all your rental passes, you can present them
with your receipt for a refund. Of course, you'd need to know this in advance. The only place to get a refund is at any of the 4 stroller rental places, one in each of the theme parks. You can't get a refund at your resort or by mail.