Oddly, no... she doesn't. I feel like she should, but Kat-Car has no tattoos. My head cannon says that Marie continually makes fun of her for that.
Kat-Car would get (may get):
-A dragon, probably a perfect replica of her darling young horntail
-Somewhere discrete, in swirly small black letters "Mia" for the mother she never appreciated enough. (@Emily: if she gets this we need to have a Thalotta scene where Thalia asks her about this "Mia" and Kat-Car doesn't wanna say anything and they fight. Kay? Kay.)
Technically, she has the Dark Mark, if that counts as a tattoo for the purposes of this question. I'm not gonna tell you where it is, other than that it's certainly not on her forearm. (She probably hasn't gotten a tattoo in the past because it would make her too recognizable. Anything she got would need to be in a place easily and often hidden).
If your character had to choose between never casting another Charm, or never Transfiguring (this includes vanishing and all else that falls under Transfig) anything again, which one would they pick?