I dont want this to turn into a debate about pre-Clovis migration. A simple tick could infect a small animal that would be the vector for disease to larger animal species.
Lets remember that a lot of infections are transmitted through water,air, foodstuffs and waste.
All these animals could carry diseases to the New World, as could the hides of these animals. Perhaps a combo of the three factors of disease,hunting and environmental change would be a more logical conclusion.