Unfortunately where the JC is concerned, there is no “just” about it.
The JC doesn’t issue duplicate papers unless you have a notarized document from the person who lost them–presumably the previous owner–detailing how they became lost (which needs to be more like “burned up in a fire” than “Put them down somewhere and can’t remember where”.) Until that happens, the JC has no proof that the horse actually belongs to you.
A TB without registration papers cannot be entered in a race. It’s not unusual for TBs to be sold or given away for a second career without their papers so that the new owner doesn’t turn around and try to take the retired horse back to the track.