Well, you need a connection to the Internet to set up the PC, so that a) it can verify the key is valid, and b) start downloading any driver/app/Windows updates that may be needed.
You don't need a Wi-Fi adapter, as long as your PC is hardwired to the router/modem with an Ethernet cable. If you don't have it hardwired in, & somehow it is physically incapable of being hardwired into the router/modem (i.e. it was mounted into the ceiling of your garage), then you'll need a Wi-Fi adapter to get the Internet connection.