This is an interesting tale. I am not sure exactly how impossible it is for a dog to immediately become a transmitter of the venom a snake just injected into it but I suspect that impossible is the key word here. Clearly these people believed it though.
I wonder if James Wilkinson actually showed real medical signs of poisoning or if he just convinced himself he was affected? I suspect that he was actually bitten by the snake while he was paying attention to the dog that was chewing on him!