First and foremost you will need a good helmet. This helmet will need to cover the temple area and ears. BMX is different from other forms of bicycle racing, in that the riders are all in close proximity and there is the danger of falling. This requirement is for your riders safety.
The only additional items are long sleeve shirt or jacket, long pants and shoes. Elbow pads, knee pads and shin guards are acceptable but do not replace the requirements for long sleeves and long pants. Gloves are recommended but not required.