The story itself never states directly what year it was. So, we have to do a bit of research, and inferring from the text, to try to guess what time period it took place during. All of the evidence seems to point to the time period of the late 1800's.
If you look up information on O. Henry, his life span was relatively short. He lived from 1862-1910. Most of his stories seem to be set in around this time period, so we can probably assume that this one was also. It was a time period where cars weren't really around too much, as they "drove in a buggy" to go kidnap the little boy. Also, if you look at the money figures; Bill and the narrator needed only $2600 to set up a "fraudulent town-lot scheme in Western Illinois." $2600 isn't very much money, but in the late 1800's, it would be quite a bit, and enough to pull off a scam. Also, in the late 1800's, Illinois was still a pretty unsettled, "Western" frontier town, and so they were headed out to start a fake town, and ask for investors to put money into it. So, the west wasn't that settled yet, which fits the time period of the late 1800's.
Also, the trio set up camp outside of town; nowadays, that wouldn't be possible as everywhere is settled, or land is owned. So, the land was still uninhabited and people could find a good place to hide in.
Using all of this evidence to support you, you could probably safely assert that the story occurred sometime in the late 1800's. I hope that these thoughts helped a bit; good luck!