It is very typical for the key of A minor to have the G# leading tone, which helps to add tension to the V (E) chord, and brings a clear resolution to the tonic (A). However, even if only G-natural is used, if your listener still feels a sense that A is the tonic, i.e., the piece revolves around A, then that would make a safe argument that the piece is indeed in A minor.
Furthermore, if it revolves around A, then there is no argument that it is in C. For a piece to be in a certain key, it needs to exhibit a sense that the first pitch of the key (the tonic), is the home, or center, of the piece. Examples of ways a piece revolves around a tonic are the final chord of the piece being the tonic chord, frequent pauses on the tonic chord at the end of phrases (i.e., cadences), etc.
Arguably, you could say that your piece is modal, and not tonal, since you're not using the leading-tone (G#) for voice leading and tension towards the tonic. But even if it's labeled modal, it should still be emphasized in your notes that it revolves around A (if that is truly the case).
A good example of a song that does this is "Smokin' Gun" by Robert Cray. He doesn't use the major V chord in his progression, but clearly the song is still in a minor key (Em).