The domestication event would have been very early... earlier than in Eurasia. That's not to say it couldn't happen, it just would have had to happen differently. There were no domestic grains or pulses yet, but they did have dogs, I believe. They would have needed to come upon the concept of herd management, I think... they probably would use the horses as food still, but exert more control over a single herd, in order to maintain a constant food source. Once they'd gotten them fenced in somehow, domestication would follow eventually. I think it is also possible for horseback riding to precede chariots. The young of the controlled herd would grow up with more familiarity with humans, and eventually (perhaps a few generations in) they would probably be willing to let someone lead them... then a few more generations, and you could be riding.
There's going to be a huge effect on everything else that happens in the Americas afterward. For one thing, having domesticated horses in 10,000 BC puts them about 6000 years ahead of Eurasia. So much for having to play catch up.