I suspect the "shoot for" isn't the relevant phrase here. It could be any verb, the relevant part is "for shit" - so you could say "he can't sing for shit". It is the same as a similar version " can't x for toffee" - where even with the incentive of toffee a person would be unable to do a thing. The suggestion is that even if a challenge (singing, shooting) was so easy that there was a prize like shit they would fail.
That said "can't shoot for shit" is a particular common version of that phrase, just because of the alliteration.