For some reason, Goat Carts appear to have been a bit of a fad in the early years of the 20th century. It was the ride-on vehicle of its time. Here is a letter from the December 23, 1902 Richmond Evening Leader from a little boy requesting a Billy Goat and wagon. It speaks to an earlier time when America was a more rural place and where even in cities there was room to maintain a goat and park your wagon.
I am not sure what caused the Goat Cart craze nor what ended it but here are some images from the time: