you can't really have a 'nonsense' name - there's a list of around 3000 characters you're allowed to use in names, but in theory, you can put together whatever you want. you'll just get side-eyed for having a weird name, that's all.

that being said, i think most of these websites that ask for a kanji name will require you to show ID with that name when you show up in person, so you might run into trouble if you just pick random characters.

There are only 3000 legal kanji characters. But people have started naming their children in hiragana and katakana. Still no romaji allowed. Yet.

You can register a legal alias (通称名) that you can use on forms. I know people who have done this with just single a single kanji to avoid a lot of the headache associated with having a foreign name or (god forbid) a middle name. I've considered a few times registering my legal alias as 一一 to have a two stroke full name.

Rather, only kanji name is legally recognized so random sequence is technically possible. But that'll just throw exceptions everywhere.