Dictionaries tend to give both. Presumably the ch spelling alludes to the [speculative] origin in French chantez, while the sh one is a more faithful rendering of seamen's pronunciation — with perhaps some anxiety, among 'chantey'-users, to avoid confusion with songs sung in rude shacks ['shanties']. There isn't much ambiguity, whichever is used; so why not just regard it, as the dictionaries do, as optional; and find some more worthwhile object of worry to eat one's ♥ out over!?