Can I get tattoos in AIT?

Google is your friend:
TRADOC Regulation 350-6

"2-8. Tattoos/brands
TRADOC IET recruit tattoo/brand policies apply to all Active Army (AA), U.S. Army Reserve
(USAR), and Army National Guard (ARNG) Trainee/Soldiers whether non-prior service or prior
service. Army policy governing tattoos is found in Army Regulation 670-1, paragraph 3-3."

So whatever it says in AR 670-1, paragraph 3-3 applies to IET soldiers too.

All I did to find this was Google the search terms "TRADOC SOP on tattoos."

Depends on what you’re going to AIT for. Do you have some MOS that will keep you riding a desk? Or will you be spending time getting down and dirty. It will make a huge difference. Also, best advice I ever heard was if you really want something as a tattoo get a T-shirt with that design on it. Wear that for a while. I think it’s pretty sound advice, but then again I have two full sleeves, chest piece, shoulders back and one leg tattooed. If I could start over with them I wouldn’t have gotten any to begin with.

I have an opinion, but you don't have to do anything I say because people will do what they want to do regardless. I've been asked that question many times before by friends. I assume reading your question again, you currently don't have tattoos; I say, if you weren't born with one, you don't need one. Again, I'm just a civilian.