I get a little nerdy about this kind of crossword clue, so here's the short, useful rundown I use when a puzzle says simply "Japanese martial art" and you're wondering which kata lines up.
If the grid wants a six-letter fill, 'KARATE' is the most common pick and its kata families are the ones you usually think of: Heian (also called Pinan), Sanchin, Bassai, Tekki, and so on. Those are the formal patterns practised solo to drill technique, balance and timing. If the puzzle is four letters, 'JUDO' is the usual fill — judo has formal kata too, like 'Nage-no-kata' (throws) or 'Katame-no-kata' (holds and submissions). For five letters, 'KENDO' or 'IAIDO' crop up and both have their own kata traditions (kendo kata emphasizes partner forms with bokken; iaido kata are the drawn-sword routines).
So, match the grid length first and then think about which kata family belongs to that art. I usually picture the kata name in my head — Sanchin makes me think of stiff breathing and rooted stances, Nage-no-kata of partner throws — and that mental image helps lock the right martial art into the puzzle. It’s a small delight when the crosses confirm it, and I always feel smug and oddly peaceful afterwards.