Can a kid use a compound bow?

Anyone with adequate training in the skill and safe usage of a bow can shoot with any bow that is right for them.

Remember that the bow should be the right "draw weight" for them, draw weight is strength and a bow that is too powerful will make it unsafe to use in the person's hands, this applies to adults as well as children.

If you get your child some training and provide them with a suitable compound bow, they will be perfectly capable and allowed to use one on a suitable Archery range.

(NOTE: I am a real archer and I am an expert on the subject)

...This is just my personal opinion and is simply worth consideration to you, but traditional archery is ideal for the sport of archery becasue you can appreciate the extra skill needed to use it and the feeling of connection to history can be well rewarding.