By regular means, you can only get one of each Apricorn Ball per game. However, there are/have been a few other methods of getting multiple of the same type of Apricorn Ball:
1) Trade from another Sun/Moon game.
2) Participate in Global Missions. So far, two Apricorn Balls have been available as rewards: the Friend Ball (Global Mission #3) and the Heavy Ball (Global Mission #4). Both events have ended, however there are more coming in the future and it makes sense for Apricorn Balls to be prizes as well.
3) So far, most generations of Pokémon have had a cloning glitch of some form, so it is possible that someone will figure out how to do it in Sun/Moon. Held items are cloned alongside the Pokémon, so that could be another possible way of getting multiple Apricorn Balls. However, there is no guarantee of a cloning method being discovered in the future.
4) It is possible to hack in items with third-party software.