No, you're not going to get tattoos as we currently (kind of) understand them, but since you opened it up to other forms of advanced technology, I'd suggest cells or protein nano-machines.
The first thing to understand is how tattoos work in mammals:
- We have skin cells that produce melanin in response to UV exposure. This prevents UV radiation from penetrating deeper into our skin.
- Over time, most of the melanin fades, but some of it is absorbed by specialized immune cells.
- These immune cells hang out in the skin with their melanin, providing some protection from UV's.
- When they die, another immune cell of the same type absorbs the melanin to keep doing the job of protecting from UV's, usually in roughly the same place as the previous cell.
Tattoos are a hack of this system. They introduce tattoo ink under the skin, and the immune cells think it's melanin and absorb it. (The life and death cycle of these cells could explain why tattoos fade and migrate over time, but that's an answer for a different question.)
You can watch more about this here:
In contrast to self-contained cells in the sin, chitin is more like a biologically-produced mineral.
Like many minerals, it has a specific porosity. It's used as a cell wall by some fungi, and this is how microbiologists discovered that they could stain it. The stain enters the "pores" of the chitin and binds to specific molecular points. This is similar to writing on your hand with a permanent marker.
The problems with microbiological stains are that they can be dissolved in the appropriate solvent, or, in living creatures, wear off over time as biological material is replaced.
So what if your crab-men reverse engineered the cells that contain the tattoo ink? So instead of needles injecting ink, the needles inject the cells or other proteins that contain the ink. This could be done while the chitin is soft, just after a molt.
The cultural questions I can imagine this leading to are, how did your crab-men discover the immune cells, and why did they adopt the practice of tattooing themselves with it?