Although you will need an internet connection to download your software and access the assembly instructions for your LongMill, for everything else, no, you do not need an internet connection or wifi to run or use your machine. All gcode sending software such as UGS and gSender which is used to control and send programs to your machine, work completely offline.
In terms of CAD, CAM, and design software in general, some which are cloud-based will need an internet connection to work (Easel, CAMLab, Onshape), whereas others do not (Carbide Create, Vectric, Carveco). You can either choose software that fits your needs based on whether it needs to be connected to the internet, or generate the programs on a computer that is connected to the internet first, and then move it using a portable storage medium like a USB stick onto the computer that controls your machine.