The fastest way is probably through Poké Pelago. Send your Pokémon into the caves of Isle Aphun, on the path "Path for Odd Shard-Hunting!" where they have a high chance of bringing back Shards. Then you can trade these Shards in for Silver Bottle Caps in the Festival Plaza. However, there is one other method that can be a good way to accumulate some caps:
Treasure Hunt ★ or ★★
"Wait, what?" I hear you say. As I mentioned above, there is a way to reliably get normal Bottlecaps, and this is how. Every time you get a new Lottery Shop in your festival plaza, you are guaranteed to get a normal Bottlecap as your first prize. New shops reset the daily timer, too. ★ and ★★ Treasure Hunt are cheap to buy, so you can keep buying this off a friend or other player and replacing your previous Treasure Hunt to get a new bottlecap each time. Buying shops does cost FC, which means in essence you are spending 150 or 200 FC for a Bottlecap, but it is at least consistent. Remember that duplicate shops cost more to buy, hence why you need to alternate between ★ and ★★ stars each time you want a new one. Check this thread here for more discussion on this mechanic.
~ Source
This method is a handy exploit to get Bottle Caps, albeit a bit tedious and you have to suffer through using Festival Plaza. That whole thread has information on how to get Bottle Caps from a few other places in the Plaza.
A piece of advice: Farming Bottle Caps takes a lot of effort, and no small amount of luck. Breeding for perfect IVs is going to be much, much quicker. Trust me, I bred a lot of Pokémon in ORAS. With a 5-6IV Ditto, I've gone from zero to five perfect IVs in under an hour. I would only recommend Bottle Caps for the Tapus, Ultra Beasts, Shinies, Ditto, and Type: Null, because you cannot breed them.
Of course, I would be remiss if I recommended to you an exceptionally rare 6IV Ditto offhandedly without telling you how to acquire one. Personally, I would recommend going to some forum and asking for one. For guaranteed results you should try r/BreedingDittos once they open up again. Smogon and Nugget Bridge are also options; just make a thread asking if anyone is willing to trade/give you a 6IV Ditto. Keep in mind however, that the game is fairly new and 6IV Dittos will be rare. They will become more common once PokéBank opens up again.
One more note: if cloning is still a thing, you can clone a Pokémon holding a Bottle Cap over and over for unending Bottle Caps.
The event Magearna also comes with a Silver Bottle Cap.
This was very wordy, but I hope it helps you!
