I've only rented one time, and it happened to be this past May. It was never so crowded that there were issues with any parks running out of strollers.
If it hasn't changed since May, here's how the length of stay rental works:
The very first park you go into, go the stroller rental area and wait in the line that ends at a cash register. Tell them how many days' rental you want. You will pay in advance for each day's rental, minus 10%. You will get a ticket (it says "Rental Pass") for each day's rental, attached in a strip. This strip will be stapled to your receipt.
You then get into the stroller pick up line. They will stamp today's date on one of the rental passes. You will get a large index card to write your name on (we picked a different character's name each day), and you take that card and place it into the little card holder window that's on each stroller.
When you return your stroller: if you're park hopping, tell them that and DON'T present them with your stamped Rental Pass. Go to the next park, get right in the stroller line (NOT the cash register line), which is MUCH faster than the pay line, show them your stamped rental pass and you're on your way. If you're done with strollers for the day, you have the option of turning in your used Rental Pass for one Disney Dollar. I held onto all my stamped Rental Passes until the end of my trip and collected some serious cash, 6 Disney Dollars!!!
The next day you go to a park, do the same as above in the park hopping directions -- get directly into the line for the strollers (NOT for the cash register), they'll stamp your next Rental Pass and you're on your way.
HOLD ONTO THOSE RENTAL PASSES!!! I can't imagine they're replacable if lost -- what's to stop someone from selling them, then trying to scam Disney to replace the "lost" ones?
I hope this all helps!
-- Eric
