Yes, there are marked "stroller parking" areas all through out WDW. They are usually easy to spot. The easiest thing to do is to mark your stroller with something to easily identify it as yours and make it stand out so that no one will mistake it for theirs. I have read here on the Dis that Stroller theft does occur occasionally, but believe that
most often, it's a case of mistaken identity, not deliberate theft. Do a search or just scan through the past couple of pages on this board and you will find lots of ideas on how to mark your stroller to make it more unique.
The CM's do manage the stroller parking areas in order to keep them orderly and make room as people come and go, so if you return from an attraction and it isn't where you left it, don't panic. Just look around carefully and you'll probably find it nearby. This is another reason to mark your stroller with a bright bandana, ribbon, balloon...you get the idea!
We have been to WDW 10x, and if you added up all the days probably at least 2-3mos worth, with three different strollers of different varieties. The lates a single jogging stroller w/swivel wheel. We have never had a stroller stolen from us. We also do not leave valuables in it when we go on rides. I have no qualms about leaving items that are easily replaced though.
As for you what you're kids will need - Take into account the sheer amount of walking one can do at WDW, the humidity, the crowd levels (strollers are great for keeping kids contained and nearby), and your personal touring style. We will be returning to WDW in a few weeks with our 5-1/2yr old and will be bringing our swivel-wheel jogging stroller for hm to ride in. I suspect he will be doing a lot more walking this trip however we wouldn't dare go w/out the stroller because I know there will be times he will be worn out (especially at the end of the day) and we will not be able to carry him very far. So for us, the stroller is a convenience item we don't really use around home any more but we kept for use at WDW.
Also, keep in mind that WDW has rental strollers as well so if you go without one but then find you do need it after all, you can always rent. Renting isn't cheap but the strollers are very sturdy. $15/day for a single, $31/day for a double and you save a couple of bucks a day if you do a multi-day rental. More info and pics of the strollers
here.
hth!