If you continue to see the error after verification, check your antivirus quarantine or reinstall the game via Steam. No external DLL download is necessary or safe.
You should not download a standalone steam_api.dll from third-party websites. That file is a proprietary Steamworks API component, and obtaining it outside of a legitimate Steam game installation is risky and often illegal. Call Of Duty Modern Warfare 2 Steam-api Dll Download
I understand you're looking for information regarding the steam_api.dll file for Call of Duty: Modern Warfare 2 on Steam. However, I must provide a responsible answer, as this topic sits in a legally and technically sensitive area. If you continue to see the error after