Most PS APIs that you would want to access programmatically are available by adding .json to the URL.
They all have Access-Control-Allow-Origin: *, so you can access them directly using AJAX.
Getting a replay:
https://replay.pokemonshowdown.com/gen8doublesubers-1097585496.json
https://replay.pokemonshowdown.com/gen8doublesubers-1097585496.log
List recent replays:
https://replay.pokemonshowdown.com/search.json
Search by user:
https://replay.pokemonshowdown.com/search.json?user=zarel
Search by multiple users:
https://replay.pokemonshowdown.com/search.json?user=zarel&user2=yuyuko
Search by format:
https://replay.pokemonshowdown.com/search.json?format=gen8ou
Combined searching:
https://replay.pokemonshowdown.com/search.json?user=zarel&user2=yuyuko&format=gen7randombattle
Paginate searches:
https://replay.pokemonshowdown.com/search.json?user=zarel&page=2
Searches are limited to 51 results, and pages are offset by 50 each, so the existence of a 51st result means that there's at least one more page available.
Pagination is not supported for the recent replays list, but is supported for everything else.
https://pokemonshowdown.com/users/zarel.json
https://pokemonshowdown.com/ladder/gen8ou.json
https://pokemonshowdown.com/news.json
https://pokemonshowdown.com/news/270.json