Hello
If you mean sending a data from Arduino(Board&Circuit) to and API , this can be done as follows
1-ESP8266 connected to Arduino OR more conveniently a Nodemcu board
2-Connection to the Router through the ESP8266 protocol(STA_Mode) and from there it's just a matter
of software to send a Json message .
If you mean the opposite , I.e you want to send from the Internet and it reaches your Arduino/other Dev Board .. You'll need a static IP For your router .
Kind Regards