"Wii Play: Modern Warfare 3" Doesn't Exist—Here's the Reality

That game doesn't exist. You're conflating two completely different things:

• Wii Play (2006) — A mini-game collection bundled with a Wii Remote. Includes shooting range, table tennis, fishing, etc.
• Call of Duty: Modern Warfare 3 (2011) — A standalone FPS released on PS3, Xbox 360, PC, and Wii (as a separate disc, not part of Wii Play).

The Wii version of MW3 is a distinct port by Treyarch, scaled down for Wii hardware. It has its own campaign and online multiplayer (via Nintendo Wi-Fi Connection, now defunct; community servers like Wiimmfi exist).

Updating Any Wii Game

If you have a legitimate Wii game that needs a patch: Connect to internet → Wii Shop Channel (defunct) → some games checked for updates on launch. Today, most patches are unavailable officially. Homebrew (Wii U vWii or hacked Wii) can apply fan patches for some titles.