A Team Skull NPC in the game — I believe it's the first city of the second island, after beating the Water trial — says that the chances of finding a Shiny Pokémon are "about 1 in 4000." In XY/ORAS the Shiny chance was 1/4096, so what the grunt says indicates that the Shiny rate has not changed significantly, or at all, in Sun/Moon.
Source: I talked to that Skull homie, yo.