we took DD last Fall right around her 1st bday. It's not as large as Cox Farms but it was good enough for a 1-yr old. I think they have camel rides too, which your son might enjoy. It's a bit cheaper than Cox too (at least for their Fall/Pumpkin Festival, not sure about now). Not sure if it's worth the drive from all the way in Springfield though. There is a small petting zoo in Reston, which might be closer to you. And of course the Zoo in DC (free but have to pay for parking).
Sofka is correct as it's really small. We took DD to Reston Zoo last year when she was 1 and she loved it. She loved feeding the animals and going on the big wagon ride.
Re: Leesburg animal park