Learn German: Wer sieht den Tätowierer das Tattoo stechen? - Who sees the tattoo artist inking the tattoo?

Because sehen (to see) is a verb of perception. Just like modal verbs, verbs of perception and motion allow you to use a second verb in the sentence. That second verb is pushed to the very end in its bare infinitive form, with no zu required.

The English translation says "inking" with an "-ing". Why doesn't German use a participle here?

German doesn't have an exact equivalent to the English continuous "-ing" form. When you want to express that you see, hear, or feel someone actively doing something, you just use the plain infinitive (like stechen) at the end of the sentence.

Why is "den Tätowierer" in the accusative case?

In this sentence structure, the person you are watching is the direct object of the perception verb. Since the subject (Wer) is the one doing the seeing, the tattoo artist is the one being seen, which requires the accusative case: den Tätowierer.

Wait, "das Tattoo" is also accusative. How do we know who is doing what if there are two accusatives?

Both are indeed accusative! den Tätowierer is the direct object of sieht, and das Tattoo is the direct object of stechen. We rely on word order and logic here: the person performing the second action comes right after the main verb, followed by the thing they are acting upon.