If you cannot verify files (e.g., you are offline or on a limited connection), you can manually extract the DLL from Steam’s cache:
GRID Autosport originally ships as a 32-bit application, so it uses steam_api.dll (not the 64-bit version). If you see references to steam_api64.dll, that is for a different game. Do not rename 64-bit DLLs to 32-bit—this will cause a "Bad image format" error.