Do nunchaku artists use under-the-arm protective pads during demonstrations for quick underarm catches? If so, I see none available anywhere.
I have been practicing nunchaku for a number of years now and I have never seen or heard of anyone using any padding like that.
Generally beginners will practice using foam nunchaku, which should provide more than enough padding and protection whilst they learn how to properly handle the weapon. Once they are more proficient in the weapon, they should have developed enough control over the nunchaku to be able to perform under-arm catches and rebounds with non-foam nunchaku without harming themselves.
Martial artists practicing the use of the nunchaku do not traditionally use any padding when catching the weapon under the arm. Control is exercised so as to not swing the weapon with great velocity at any part of the practitioner's own body. This learned control is one of the primary reasons why students of the nunchaku do not start their technical development using the weapon at full speed. Speed is something one works up to. It's also worth noting that most nunchaku weighted for martial arts use are not optimized for use as weapons and aren't going to do much damage without follow-through in a given strike. You might get a few bumps and bruises, but you aren't likely to break any of your own bones during practice.