Words that mean “write” usually originate as a meaning shift of an already existing word that meant “scratch”, “draw”, “carve”, or the like.

This is also the case with “write”, which is originally from the Proto-Indo-European root that meant “rip” or “tear”, which became “carve”, and then finally “write” in late Proto-Germanic/early Old English.

The only straightforward ways I can think of to get a brand new word meaning “write” after the fact would be to either: 1) coin it somehow, possibly by compounding existing morphemes or 2) borrow it directly from a language whose people already practiced writing. I’m not aware of any examples of this, but even though they are likely rare they probably do exist.