How to send data from running Webhook to Arduino board?

Hello guys, this is my first post on the forum, although I have quite an experience with Arduino projects.

On the project I am working now I have an third party app that run Webhook on given action. Some how I need to notify the Arduino board that the action has happened.

With the Arduino board there will always be a phone with mobile internet.
I can connect the board with WiFi to the phone`s hotspot with ESP8266 ESP-01 for example?
Or with Bluetooth using HC-05?
Or maybe run the board as a server with public IP?

I was thinking to use IFTTT applet, but it doesn't solve my problem end to end.

Ethernet module is not an option, because the project must be portable.

If you have some ideas, I will love to read them

Do you understand what webhooks are?

Behind a tethering smartphone? I get the impression you have no clue about networks. What exactly do you want to achieve? What exact app does have that Webhooks? We may be able to help you but only if you provide much more details.