I don't believe that any horse that is the product of artificial insemination would be considered a thoroughbred by the Jockey Club, and so would not be eligible to race at any US Thoroughbred Racetrack, and probably not at any Thoroughbred Racetrack worldwide.
ML/NJ
Artificial insemination is widely used in horse and other animal breeding.
The Jockey Club may issue a ‘racing permit’ for non-thoroughbreds that is accepted by state racing commissions.
I am sure they are very selective in granting this but I don’t know the details.