With All the Prohibitions of the Watchtower Society, Do They Have Any Teachings About Tattoos?

Actually Jehovah's organization hasn't any 'prohibitions'.

What we do or do not do, comes from God's Word.

If Jehovah tells us to 'stop touching the unclean thing' we don't.
If Jehovah tells us to abstain from something, that is what we do.

As to tattoos, having tattoos were wrong according to the Law of Moses.

But since Christians aren't under the law, then it isn't a sin to get a tattoo, unless it promotes something against God's word.

However, do tattoos identify a person with the world? 
Would a brother or sister be stumbled if we get a tattoo? 

These are a couple of questions a Witness needs to answer before they use their Christian freedom and get a tattoo, buying a certain piece of clothing or as Paul says, eating meat.
.

"Do not put tattoo marks on yourselves. I AM THE LORD" (Leviticus 19:28)!

Here is one of the Mosaic laws that wasn't fulfilled under the New Covenant.

I'm telling you; If God doesn't "put His law in your heart and mind" (Jeremiah 31:33-34; Hebrews 8:10-12; 10:16-17) after Holy Spirit baptism (Acts 1:5;8) and "gives commandments" (Acts 1:2) there is NO WAY you'll remember what laws under the Old Covenant are still effective.

GOD doesn't forget His laws; but you will (James 2:10)!

Yes. Read here: https://www.jw.org/finder?wtlocale=E&docid=502013252&srcid=share

They appear to be evangelists who would be very understanding about tattoos or any other special features of possible converts.

Well, that took very nearly 7 seconds to Google:
https://www.jw.org/en/bible-teachings/teenagers/ask/should-i-get-a-tattoo/

They don't ban tattoos, but they'd really rather you didn't have them.