My other half is a tattoo artist. What I know about getting a tattoo is:
1. In the state of TX, you MUST be 18 in order to get a tattoo. Even with parental permission, you cannot be tattooed in a licensed shop if you are under the age of 18. The ONLY exception to that is to cover up gang-related tattoos, and I believe even then you must have a court order.
2. A reputable shop will not only ask to see your ID, they will make a copy of it. The shop he works at has a contract-style form with a hole cut out for ID. They make a copy of the whole thing, and then have you sign below your copied ID. I don't know if signing a form is required for all shops, but it's a practice that all the shops I know of follow.
3. Tattoo artists are not individually licensed. Generally what happens is that a shop owner obtains a license to operate, and the employees that work in the shop work under that license as sort of an umbrella license. This means that most shop owners are EXTREMELY interested in making sure their employees toe the legal line with regards to age limits, tattoos given to people obviously under the influence, etc.
That being said, most likely one of two things happened - either he has a fairly good fake ID, or it was done at home. If it was done at home, it's unlikely that you'll ever discover who did it (although keep an ear to the ground...if it's someone your son is around often, or someone known to his group of friends, someone's bound to let something slip sometime). If he had a fake ID, then I don't know whether or not a shop could be held responsible...they may be artists, but they are not extensively trained on how to spot a fake ID. I would go ahead and contact the police...they might not be able to do very much legally, but they might could put a bit of the fear of God into your son where the legal ramifications of his actions are concerned. If nothing else they might could answer some more of your questions, and might even know names of people known to be involved in such things. If one of them rings a bell you can always pursue it.
On another vein, perhaps you could go to a reputable shop and have one of the artists talk to him about what he is doing. Most of the artists I know feel pretty strongly about tattoos being done the right way, and would probably be willing to talk to him. Perhaps he would listen to someone that's already walked the path, and has the "marks" to prove it. If you think this might be something you would be interested in, I know some artists in and around the Houston area that might be willing to do a favor. Feel free to message me.
However, this seems like a deeper issue than a bit of ink more or less. Try talking to your son a bit about what makes this so attractive to him. Be prepared to talk with an open mind...he might surprise you. You might not change his mind, but perhaps he and you could come to an agreement you both could live with, where he could make those choices responsibly. You are still the boss of your house, but it'll be a lot easier if you are both on the same team.
Best of luck to you,
-M.