One of the main issues would be the colour fading. As the kid's arm gets bigger, the skin cells that had ink injected would no longer all be adjacent, as new skin cells grow between the older cells. It wouldn't distort so much as fade somewhat. Look at photos of yourself as a kid, look at moles, scars, birth marks, etc on your arms. As your arm grows, it's size and location remains proportional to your arm.
You also need to consider what is being tattooed, the human brain is great at ignoring things, so if it is an image, then any slight distortion would mostly go unnoticed, distortion in letters or numbers however would be more noticeable.
As to the idea you could tattoo a mystery image on a kid and it would morph into a recognisable image by the time they stopped growing. Nope, look at images of human development, a kids arm is just a smaller version of an adult. You couldn't tattoo fine enough lines, close enough together to get the effect you are after.
Try drawing a large and complex design on a deflated balloon with a medium tipped sharpie. Then inflate the balloon. The this would give the effect of the kid growing up. The lines would be there, but faded as the latex is stretched.
One last thought, have you tried to get a five year old to sit still for a haircut? Imagine the fun you would have getting then to sit still for a couple of hours whilst you inject them thousands of times. Whilst I happily went back for more tattoos after my first, I seriously doubt you could get anything like straight lines on a squirming kid after the tattooist does the first line.
As to the legality of tattooing kids, whilst previous posters are right, it's illegal pretty much everywhere, there are people who will do anything if the money is right. Also look at what we do to kids in the name of religion and fashion. Many cultures mutilate infant's genitalia with circumcision (both male and female) and ear piercing is common all over the world. Depending on the society your novel is set in, tattooing kids might be normal, and when their tattoo becomes recognisable, is a mark of their arrival at adulthood.
