The Xbox 360 Wireless Network Adapter (black "N" or white "G" model) functions purely as a wireless network client. It connects your console to a local Wi-Fi network (LAN). Internet (WAN) is not required for the adapter to function.
Note: The adapter draws power from the console's rear USB port. Ensure the port isn't current-limited (some early 360 models struggle with high-draw adapters + HDD simultaneously).