Question:

Can show horses breed during their careers? How about racehorses?

Answer:

Racehorses cannot breed while actively racing. There are absolutely no exceptions to this rule. Once a racing stallion covers a mare, he will never be permitted to race again. The same goes for a racing mare: once covered by a stallion, she cannot return to racing.
Show horses, on the other hand, are permitted to breed during their careers, within reason. Stallions obviously should not be bred excessively, and if mares are bred without the use of Embryo Transfer, they must take the appropriate time off from the show ring.