Well yes, a piercing or tattoo can cause infection. If the area right under/around the tattoo and piercing are swollen, warm, itchy, leaking discharge, etc, you should go back to the artist and see what they say.
If they don't help, go to a doctor.
But what exactly did you get pierced? Your clit or your labia or something else I haven't seen before? And I'm guessing it's your genital area that's tattooed, not your vagina- since that's in your body.
