Ask the tattooist what brand ink they use, then google and email the manufacturers. That's probably the easiest way of finding out. Unless the tattooist is vegan, or has taken the liberty to look into such a thing, they probably won't know the answer themselves. The manufacturers are also not obliged to divulge full ingredients (this is what I can ascertain from looking at US based sites - maybe regulations are different in NZ??).
I was reading about this awhile ago and from what I can ascertain there are two potentially non vegan components of the ink:
The black pigment in some brands is from charcoal made from animal bones, this is usually called 'bone black'.
Some (not all) tattoo inks use glycerin as the pigment carrier - some brands may use animal derived glycerin.
Here's some info on the inks' composition - again, no idea how accurate / exhaustive this info is..http://chemistry.about.com/library/weekly/aa121602a.htm