My tattoos are done in Starbrite ink (black, but I'm almost positive they make coloured ink as well.) I visited two different parlors and they both used Starbrite without even knowing it was vegan. My boyfriend's parlor also has Starbrite ink. Call around and ask what inks the parlor uses, and if it is available in the colours you want.
Also ask about their age policy. I'm not going to tell you you're too young, since I got my first tattoo at 16 and still love it. I will tell you that legally you will probably have to be 16 with parental consent. My mom wasn't present when I first got my tattoo, but she did come with me to make the appointment. We needed my ID, her ID, my birth certificate, and a notarized consent form. When my little sister wanted to get her tattoo, we were informed that the policy had since changed and a parent would have to be present for the tattooing. If your guardian will not consent to you getting this tattoo, you'll have to wait until you are 18. Any place that will tattoo a minor without the guardian's consent is breaking the law, along with who knows how many other laws. Broken laws in a tattoo parlor usually mean health risks.