tattoo printer

There are some, but there are several reasons that you haven't seen them about.

The people who spend the most on tattoos are doing so because they are art made by an artist. I'm not sure if you know many people in the scene, but these aren't the people getting flash tattoos at a random shop. They plan their tattoos, they pick their artists, and they spend many hours trusting another human to permanently alter their body.
The most common type of printers in the world print on a 2D plane, even most 3D printers are printing on a nearly flat 2D plane one layer at a time. The human body is largely not a 2D plane.
Human skin varies, ask any tattoo artist. You can't use the same pressure or the same type of needle on every person. And it takes people with experience to identify this. People who pay the most for their tattoos expect this expertise.

Those last two reasons are not impossible solves in any sense, but they do greatly increase the cost and complexity of a machine that can automatically tattoo a person without injuring them.

I'll also throw a number 4 in here. Speaking for the US, unless you've made a name for yourself tattoo artists are largely exploited here. They are mostly misclassified as independent contractors but then treated as employees. They are under valued and under paid.

So, how inexpensive can you make a machine to both purchase and maintain, while also being easy enough to undercut the already exploited labor in your average tattoo shop?

I am super proud of my plethora of bad tattoos, because they were all done freehand by the artist based on minimal input from me.
- Just think of how much better it would have been if you could have drawn something yourself, or just picked some shit off the internet and then jammed a random body part into a machine.
Yeah, a tattoo printer would have to be at least a 5 axis robot. Technically, that's not a huge issue, and even pressure sensing or using machine vision to adjust the print aren't that difficult to do.
But even if it becomes a mass produced device, manufacturing costs for the robot part alone would be at least 3k-5k and then you will need a skilled operator to control that thing.
So you are replacing a minimum wage tattoo artist with an expensive robot and an even more expensive robot operator.
Doesn't really make sense.
The same thing holds true, btw, for pretty much all "humanoid robot applications". Minimum wage wokers are incredibly cheap, maintenance and setup costs (aka healthcare and education) are paid for by the employee, there's no vendor lock in and if you don't need them anymore you can just fire them whenever you want.
That's incredibly hard to compete against for a generalized robot.
I'll throw in another point, people are fucking terrible st staying still

https://blackdot.tattoo/#introduction

It does.

I think I've worked in automation long enough to feel super uncomfortable with the idea of a tattoo print machine being anywhere near my body.
Even if I had a kill switch in hand, it still makes me uncomfortable. In general machines don't care about fleshy bits at all. If something happens, for example a sensor ages and becomes defective, the printer has the potential to cause serious harm.
I probably also hold a bit of bias, I prefer the imperfections of human, hand made art over digitized perfection from machines.
- Personally I would never get an automated tattoo, as you said the risk of serious harm is way too high, and I also prefer to keep art in the realm of Humans. I think eventually we will realize that automation isn't the answer for everything, but it is interesting that someone thought this was a good idea.
thats intresting... the site crashed my phone lll
- I clicked the link and immediately thought "wow this is going to be nothing more than an obnoxious load of marketing wank" as soon as I saw the loading bar appear and take 20 seconds to fill.
  I was right.
- Yeah, the site doesn't seem very mobile friendly. Sorry about that. But it was an interesting find, I didn't realize something like this existed until I saw your question.
"exist" is a pretty strong word for this
- Feel free to argue semantics with someone who cares.

With the lack of it existing comercial, why not make your own from a old printer like Emily the engineer did with a old 3d printer (youtube link)

What could possibly go wrong?

Fun watch

The YouTuber Emily the Engineer made one and tested it on her assistant, with success. But the video wouldn't make you want to try it.