I get asked this question all the time simply because I don't have any visible tattoos unless i wear a pair of pumps. The other one being "do you have to tattoo yourself?"
I think it's a reasonable comment that no you don't have to be tattooed. But you can probably bet your bottom dollar that you will surely get some funny looks because "tattooed folk" tend to trust "other tattooed folk" both under the needle and on the desk. Maybe being in a studio so young might favorably make you think twice about what you get.
I would find a heavily tattooed teenager on a desk quite intimidating or just wrong to look at. It can scream scene.
Having no tattoos won't make your art any less good/great or make your standard of work drop. In fact, i found that older women that were often put off from tattooing have now started to come into the shop more and more. To make you feel better, i don't consider myself tattooed very much. Nothing above the waist at least. So I often get asked if i have any at all.
It's a bit of a double-edged sword i'm afraid but hang in there