As always, I can only answer re: Ricean vampires, I’m not aware of any other vampires in other media having tatts… maybe in Twilight they can/do?
Definitively, it’s not addressed in VC canon, I don’t remember any of the vampires having tatts (whether they got them before or after they were turned) so it’s open to interpretation, #Your headcanon may vary. Personally, I have a bunch of thoughts on this mixed into my #tattoo tag, so check that out.
My headcanon is that vampires can get a tatt (and it will look permanent) but it will fade during the deathsleep, which is what @frankenland informed us that Anne Rice thinks about it, too:

If we accept the author’s words as canon, we can say definitively:
1) Tattoos that the vampire tries to get AFTER they have been turned: Will vanish in 24 hrs.
2) Tattoos that the vampire has BEFORE they have been turned: Will fade/lighten/change (see comment from @monstersinthecosmos on my post about this, below)
@monstersinthecosmos added:
i remember Anne said on FB once that someone’s tattoos would probably go really light and lacey/elegant looking after they’re turned, but remain in a way that looks ~ethereal~ lolol
^I’d have to find the post but I would imagine that AR means that the tattoo will “go really light and lacey/elegant looking” over time, not at the moment of turning, but it could be interpreted either way.
@thebibliosphere added to one of my posts: “I treat vampires as just very long living humans, which means even tattoos fade over time as the skin cells regenerate.” [X] I’m inclined to agree. I also think a tattoo could be removed from vampire flesh “by scraping/cutting/etc. off the tattooed skin and letting it regenerate to its natural state).” as @skeletalroses put it [X].