Can I get a tattoo on holiday block leave?

Time for the difference between “can” and “should”.

“Can you get a tattoo during block leave?” Yes, pretty sure you could.

“Should you get a tattoo during block leave?” Yeah, pretty sure you shouldn’t. There’s aftercare instructions that you probably won’t be able to follow during AIT in January.

Could and should are two different things.
While attending training you could run into issues with keeping it clean and follow-up maintenance to prevent damage, scarring or infection.
I am also guessing it may be an issue in regards to social distancing requirements you may have.
I would just sit on it for the time being. You have plenty of time to plan out your ink and get it done during a better time that may be more appropriate and feasible. Do some research and find a quality place with a quality design. Most of us have a jacked up tattoo from entry level training that has either been removed or had to be fixed down the line. Don't rush it.

Tattoos are a sensitive topic. They are commonly used to disqualify from re-enlistment should the army feel the need to shrink it's numbers. So make sure you don't cross the line of what is and is not permitted. Also, consider what is permitted today might not be permitted tomorrow. A non tattoo will never be prohibited.

It's also pretty clear evidence of breaking CV-19 distancing requirements should you contract anything because you went into a parlor that jabbed needles into you.

There is plenty of time to get that stuff done. I get it, you're young. The bigger prize however is getting through TRADOC and being a MOSQ soldier. Once you get that you are a little safer in regard to pushing the limits. Right now, if something happened that caused a disruption to your training you are in a prime position for the Army to just wash their hands of you.