Which board?
You can put a simple webserver on an ESP8266 board. If the user and the ESP are on the same LAN (same router), then your idea of a remote button will work just fine.
In my case, I plan to trigger the vend with an MQTT command over WiFi. This way I can make various sensors trigger the vend.
How did you form the delivery spring and how did you connect it to the servo motor?