There is no rule against carrying, or appropriately using, a whip or crop. But there are potential penalties for abusive use of the whip. And using the whip will definitely count against your placing.
Abuse is defined in GR839 Cruelty to and Abuse of a Horse. With regard to use of the whip it says
“Excessive use of a whip on any horse in a stall, runway, schooling area, competition ring or elsewhere on the com-petition grounds, before or during a competition, by any person. Except in emergency situations, any striking of the horse’s head (on the poll and forward of the poll) with the whip shall be deemed excessive.”
Penalties include “official warning, elimination, or other sanctions which may be deemed appropriate by the Show Committee. Such action(s) could include, but are not limited to excessive use of the whip or spurs.”
The Hunter rules, HU100, say
“d. Excessive use of a whip on any horse in a stall, aisle, schooling area, competition ring or elsewhere on the com-petition grounds. The use of a whip must be for a good reason, done at the appropriate time, in the proper place and with appropriate restraint. Excessive whipping will not be tolerated.
e. Any striking of the horse’s head (on the poll or forward of the poll) shall be deemed excessive”
and
“Competitors found to be in violation of these rules by the Competition Steward or other Federation Official shall be eliminated/disqualified from competing within the upcoming 24-hour period and shall forfeit all entry fees for such competition. In addition, after consultation between Competition Management, the Steward, or other official involved, and a Senior Judge in the division, additional penalties including disqualifi-cation from the balance of the competition may be imposed. All such violations must be recorded in the Steward’s Report and, if cruelty or abusive behavior is evident, it will be reviewed by the Federation for such action as may be deemed appropriate to the particular circumstances.”
The Equitation division has virtually the same rule.
In Eventing, EV111.3 says
“WHIP. The use of the whip must be for a good reason, at an appropriate time, in the right place, and with appropri-ate severity.
a. Reason—the whip must only be used either as an aid to encourage the horse forward, or as a reprimand. It must never be used to vent a rider’s temper. Such use is always excessive.
b. Time—As an aid, the only appropriate time is when a horse is reluctant to go forward under normal aids of the seat and legs. As a reprimand, the only appropriate time is immediately after a horse has been disobedient, e.g. napping or refusing. The whip should not be used after elimination. The whip should not be used after a horse has jumped the last fence on a course. The whip should not be used multiple times between fences.
c. Place—As an aid to go forward, the whip may be used down the shoulder or behind the rider’s leg. As a repri-mand, it must only be used behind the rider’s leg. It must never be used overhand, e.g. a whip in the right hand being used on the left flank. The use of a whip on a horse’s head, neck, etc., is always excessive use.
d. Severity—As a reprimand only, a horse may be hit hard. However, it should never be hit more than two times for any one incident. If a horse’s skin is broken or has visible marks the use of the whip will always be deemed to be excessive”
There was an attempt, a couple of years ago, to incorporate similar wording into the General rules, which was voted down, largely, IIRC, because “after a horse has been disobedient” is less well defined in the non-jumping disciplines.
This rule does NOT apply to Hunter classes. None the less, it provides some guidance on what could be considered “abusive” or “excessive”.