Note that I don't have ZA and have no experience myself but here are some methods I've seen in a video and online.
Basically, shinies don't despawn unless you kill them (I think) or they get scared and run away. The game can store up to ten shinies. This is important.
In zone 3, there's a building where if you hold down at a stairway to cycle between the rooftop (with the alpha fletching) and the bottom (with the alpha litleo), you can quickly despawn a bunch of Pokemon until you get a shiny in any of the nearby spawns. Apparently there is a radius so the spawns for this are quite limited.
By constantly flying to the wild zones, you can quickly despawn and spawn a bunch of Pokemon. If you do this for a long time (40-50 minutes, a couple hours), you can go in and check for shinies. Multiple can appear at once. I think you need to be careful with the day/night cycle transition when in the wild zone but I'm not sure.
Similar to the building one, if you have a pro controller, you can spam interact with the benches to spawn and despawn a bunch of Pokemon.
These methods, particularly 1 and 3, are really afk/doing one thing until something finally appears, they may take a long time, and you're definitely going to want to rely on the shiny sound to navigate to your shinies. Another flaw is that I don't think you can really guarantee a lot of targets and I'm not sure if the zone 3 method has more spots in the game where you can do it at. Again, I don't have ZA, so some of these details might be wrong or there may be a lot more methods I missed. So hopefully, you will be able to get a shiny in the meantime until someone posts a real answer.
And remember, search everywhere; the trees, the lampposts, and the rooftops. Happy hunting!
