The short answer is normally no. The long answer is if they have started in a race and if they are granted the change and pay the fee they must race under both names 3x before it completely changes. There is a rule that allows name changes for a foal. Say I pick up Sour Pickle at auction as a yearling I could probably change his name to something sweeter with that exception. A foal's name may be changed at any time prior to starting in its first race.
D. A foal's name may be changed at any time prior to starting in its first race. Ordinarily, no name change will be permitted after a horse has started in its first race or has been used for breeding purposes. However, in the event a name must be changed after a horse has started in its first race, both the old and new names should be used until the horse has raced three times following the name change. The prescribed fee (see Fee Schedule) and the Certificate of Foal Registration must accompany any request to the Registry Office for a change of name.