It "can", and if you have linguistics, its penmanship probably won't even be that bad, but it would be really tiny writing. If you expect it to write in letters sized for you, then I would expect the process to take a really long time. Even without writing you could probably communicate in other ways though, like the left pincer could be "no" and the "right" pincer could be yes.
https://aonprd.com/MonsterDisplay.aspx?ItemName=House%20Centipede
Why would the writing need to be tiny?
IMO chicken scratch writing is probably about right as a centipede is not really equipped to use a writing implement well. If I was your DM that's what I would go with. I suppose you could make a magic pen...
The writing is tiny, because the creature is tiny. At biggest, it's the size of a house cat and it's probably smaller than that. A real house centipede is barely as big as a mouse, but it's probably not that small, or it'd be diminutive size. Either way, it'd be like us trying to make letters for a t-rex to read.
I assumed the creature would just dip its feet in the ink and use that to do its writing.