And arrow becomes a projectile when shot.
And you can attach explosions to projectiles which will allow an enchantment on it which will allow a magic effect which can have a script on it.
So yes, you can put a script on a shot arrow.
In my levelers Tower mod (link in signature line) I have 3 arrows that explode on impact.
To look at the 'explosion' to 'script' path I noted above, check out the "LevelersixplosionArenaDeath01" explosion and follow it through to the script.