No,
stroller rentals are not available at Downtown Disney like they are at the parks. At least, to my knowledge.
As a side note, I have never understood why anyone would want to rent the park strollers. You have to wait in (yet another) line to check it out, and if you park-hop, it has to be returned before leaving. Then at the new park arrival, you have to wait in (yet another) line to check it out, if there are any left to check out.
I do not like to travel with our stroller; just one more thing to pack/lug/schlep/airline-baggage-check-and-hope-they-don't-destroy-it, etc. However, I have used (with much satisfaction and happiness)
A Baby's Best Friend in Orlando that has nice clean, comfy strollers (and other things), and at a better price than if I had rented at the parks. Additionally, it is with me whenever I want, as I have it my entire stay. They deliver to our hotel before we get there, (luggage room), and pick it up after we check out. It's great, and I wouldn't do Disney World any other way.
Sorry to digress; just my thought. Their web is
www.abbf.com
Hope this helps.