You are right, there is no such API - typically one adds a chess library to the application to do things like that.
Here's an example in Java, with an HTTP API, https://gist.github.com/tors42/cf4def84a0b4cb6f3410fce3b6e4ff94
$ curl --get --data-urlencode 'uci=e2e4 e7e5 b1c3' 'http://127.0.0.1:1234/san'
e4 e5 Nc3
$ curl --get --data-urlencode 'uci=e2e4 e7e5 b1c3' 'http://127.0.0.1:1234/fen'
rnbqkbnr/pppp1ppp/8/4p3/4P3/2N5/PPPP1PPP/R1BQKBNR b KQkq - 1 2
$ curl --get --data-urlencode 'uci=e2e4 e7e5 b1c3 g8f6 f1c4' 'http://127.0.0.1:1234/pgn'
1. e4 e5 2. Nc3 Nf6 3. Bc4 *
$ curl --get --data-urlencode 'uci=g8f6' --data-urlencode 'fen=rnbqkbnr/pppp1ppp/8/4p3/4P3/2N5/PPPP1PPP/R1BQKBNR b KQkq - 1 2' 'http://127.0.0.1:1234/pgn'
2... Nf6 *