is someone responcible for the vet bill if they cause damage to your horse they rent

Assuming nothing to the contrary in the rental agreement, the renter could be responsible for all damages you suffer as a direct result of their negligence.  It also could be considered a breach of the contract for the renter to have failed to take proper care of the horse while the horse was in his/her care. If it was negligent for the renter to race the horse, and you can prove that the injury was caused by the race and wouldn't have occurred otherwise, you may be able to recover the money you paid for the care of the horse's injury, and possibly any rental income you lost while the horse was out of service, and diminution of the horse's value due to any continuing problems with the injury.