What need:
- WiFi connection as a client to the access point
- Receiving of the data structure from the Arduino, parsing and reliable (with repeats and error handling if required) transfer of the data to web server in GET or JSON format
- Reliable receiving and parsing data from a Web server in JSON format and transfer as data structure to the Arduino
I.e. of a kind analogue library RadioHead for RF transmitters.
Recommend, please, ready ESP8266 firmware.