Is it legal for me to let my 6 year old shoot a gun at a range?

As long as he has proper training. But make sure you are there to prevent an accident. A six year old is not really mature enough to handle a gun.

Legal-yes; wise...probably not. Look at the range rules, follow them carefully and consider a Red Ryder BB gun instead...for Christmas...when he is ten

No, it is not illegal for your child to shoot a gun at the gun range and to my knowledge, there are no age requirements for shooting a gun. So long as your child obeys the range rules and practices gun safety, you should be fine!

One thought to add, since it is tagged family law. Yes, it is presumably legal for you to allow your 6 year old to shoot. However, if you are in pending family law litigation, I would strongly recommend you consider not shooting. I myself shoot, etc., and it is an enjoyable hobby, but I would suspect that a judge sitting in custody hearing, depending on their background, familiarity with firearms, etc., might be very skeptical of a parent who takes a 6 year old to shoot anything stronger than a BB or pellet gun, and even then.... This post is also tagged convicted felon, so if you have a felony background, etc., I would hesitate even more, because felony conviction + letting 6 year old shoot (even properly supervised, etc.) is not something that is going to convince a judge you are keeping the best interest of the child in mind. Not what you'd want to hear, but this could have a reasonably strong impact on a family law case and is easily preventable, which is what I'd recommend in this case.