Can I use EyeTV W without a 3G or internet connection?

Yes.  EyeTV W sends a live TV signal straight to your iOS or Android device, using a built-in Wi-Fi transmitter, no Internet connection required

An internet connection is not required to receive TV signals.  The DTT (Freeview or DVB-T) signals are solely received via the antenna/aerial attached to the EyeTV W.

The minimal configuration to use EyeTV W is that it needs to be connected to your device using a closed wireless (Wi-Fi) network.

Wi-Fi access to the internet will not be available when EyeTV W is connected to your device via Wi-Fi.  If your device offers cellular or Bluetooth connections to the internet, then those will not be disrupted while EyeTV W is in use.