Scripture does not directly prohibit a Christian from receiving a tattoo. Some may counter this by pointing out that Leviticus 19:28 says: ‘You shall not make any cuts in your body for the dead nor make any tattoo marks on yourselves: I am the LORD'.
However this apparent prohibition is intended to primarily present the rules and practice of worship for Jews under the Old Covenant system. Christians are not required to keep the precepts of Levitical Law, for Jesus Christ is our saving substitute and as such, he has kept them for us
Furthermore, it's cherry picking scripture and as such it's nothing short of poor hermeneutics (Bible interpretation) because it seeks to inconsistently apply only a handful of Levitical Law while completely ignoring others. For example, in the same chapter, verse 19 prohibits wearing garments with two different types of materials (think poly/cotton blends) and verse 27 prohibits trimming sideburns or beards. We could go further, but hopefully you get the picture.
The truth is that tattoos in general fall under the matter of Christian liberty and conscience much like the consumption of alcohol does. However that does not give us a free pass. Check out this article on Ligonier that discusses Christian Liberty further.