The volunteer coordinator or the TD will probably walk you through what you need to do. Don’t worry it is easy.
These show horses are pretty used to it.
After the rider is done with their test, you just ask them if you can do a bit check. You should be given rubber gloves to change for each bit check. Recently, due to health concerns- bit checks have been visual. so you are check for the correct metal, correct type of bit/thickness. Other areas with no health issues- you can approach from the horse at the side, lift the lip to check it. There is a great USEF PDF document online that shows you illegal/lega/ equipment. TD/show manager may also ask you to check riders’ equipment- such as correct type of cavesson, spurs, even whip length.
Depending on the size of the class, you may just check a few per class. Double check though, it has been a while since I’ve done it.
Don’t be afraid to as the TD to help you and guide you through the first couple of bit checks- that is what they are there to do!