Unless you're one of the first in the gates, you're going to be waiting in line for the stroller..
However, as the PP mentioned, you can pay for the number of days you need a stroller, and therefore only wait in the "pay" line one time...then the other times you go to the person who hands out the strollers, hand over that day's voucher, and they give you a stroller. It does save time!
When we have rented, we like getting them in the morning, because it gets us to our morning-goals faster. DS is a bit dawdly in the mornings, but I know he'll enjoy getting to the rides we want to go on, so moving him along in the stroller is better than trying to drag him along.

If your kidlets are different, or if your plans/goals in the mornings are different, as long as it isn't a crazy-busy day, you could probably just get it later on.
About rentals being taken...most of the time (not all, but most) a missing stroller is a stroller that's been moved somewhere else by CMs for crowd-flow issues. Usually it is found right around the area. We found our stroller in Adventureland in a newly-created "corral" when we'd parked it near Jungle Cruise...I found our stroller in a different area near Pooh when we'd left it near the Canoes...and then one time...the stroller was just GONE. But the nice thing with rentals is that you can just show your receipt and get another! I never leave things in the strollers anymore (we've had things stolen more than once, including a balloon having been undone) so it's not a huge deal.