Yes, you can take the strollers into the stores. You shouldn't have any problem with an umbrella stroller. I don't think I'd want to take in a big rental stroller though. Actually my (swivel wheel) jogging stroller is just as big as a rental. We used it on our last trip and I tried to NOT take it into the stores because it's too big to maneuver thru smaller spaces. While at DTD, I left it outside a few times...with trepidation since I know stroller theft is a bigger problem there (and this isn't a cheap stroller). I tried to stay near the front of the store where I could keep an eye on it.
There are no actual restrictions though, if that's what you're wondering.