Even though almost everyone uses a chain it is not required.
Here is a quote from the APHA Rule Book 2. A clean well fitting halter with an appropriate lead is required.
Here are two quotes from the AQHA Rule Book
•tack should fit properly and be neat, clean and in good repair.
It is preferable that the exhibitor’s hand not be on the
snap or chain portion of the lead continuously.
A snap is directly in AQHA's wording. If you don't want to use a chain you will need to find a leather lead shank with chicago screws in it so you can remove the chain and replace it with a swivel snap. I had a friend that showed for years without a chain because her mare wouldn't tolerate it, she always did very well and earned many showmanship awards at breed shows.
Someone stole my %&#*@&% Carrot!