To program an AVR Arduino from the esp866 Arduino, you can use the ESP8266AVRISP library. It is bundled with the esp8266 Arduino core. It has an example.
Also an Arduino Mega can do a self update with my ArduinoOTA library and among other options for the transport of the updated binary the ArduinoOTA library can use my WiFiEspAT library which works with an ESP8266 or ESP32 with AT firmware.
An esp8266 can also be used as direct network proxy for avrdude, where avrdude works over a network port communicating with a simple network to Serial sketch running on the esp8266.