No, your smart meter doesn’t need an internet connection to work. They operate using a wireless network, but one which is separate from your home Wi-Fi.
The two wireless networks your smart meter will use are called HAN (home area network) and WAN (wide area network). Smart meters use HAN to communicate with each other and your in-home display. WAN is used when smart meters need to communicate with an energy provider.
WAN is a secure network, similar to the one your mobile phone uses to send and receive data, while HAN is a separate secure network contained within your own home.