I am looking to make a project with an ATMega328PB microcontroller. I want to tell when my mailbox has been opened and to send some sort of notification. I was thinking of use a GSM module to send this text. Is this possible and is there any other way to send some sort of notification without having to use a SIM card?
One way or another you will need some sort of communication module. You could use a GSM module as you mentioned, or a WiFi/Ethernet chip. If the latter, you can use any internet-based SMS service (Google Voice, VOIP, etc). If you go with the WiFi approach, I'd suggest an Espressif chip (ESP32 or ESP8266). They're cheap and have a lot of Arduino support.