You cannot simply download the "latest" Play Store APK found on the web, as modern versions require much newer Android builds (typically Android 6.0 or higher). To get the Play Store working on Android 4.0.3, you must target the specific versions released before the 2018 cutoff:

Avoid "modded" or unknown APK sites that may bundle malware with old software. Stick to reputable repositories that verify cryptographic signatures to ensure the files are original and untampered:

Google Play Store APK Download for Android 4.0.3: A Complete Guide