No. The Xbox 360 Wireless Network Adapter (or built-in Wi-Fi on S/E models) only connects the console to your home Wi-Fi. It does not require a separate card.
However, to play online multiplayer, access Netflix/YouTube apps, or use Games with Gold, you need an Xbox Live Gold subscription (now called Xbox Game Pass Core). The adapter is the "bridge"; Gold is the "key."
Note: As of 2024, Xbox 360 online services are still active, but Microsoft has stopped manufacturing hardware. Free-to-play games (e.g., Fortnite, Apex Legends) no longer require Gold/Core on any Xbox.