You should be able to install Plants v Zombies, the versions of of the app that are in the store only require iOS 4.3+ or 5.1.1 (Plants v Zombies 2). What happens when you try to install that app ? It is iOS 5.1.1 that you have on the iPad : Settings > General > About > Version ?
For other apps, if the current version of them requires iOS 6+ (and that is the version that you are trying to install) then you won't be able to install them. Some developers allow you to download older compatible versions of their apps, so if the app is already in your account's purchase history then try going to the Purchased tab in the App Store app on the iPad and see if you can download an older version.
This page may also help you find apps/games in the store that require iOS 5.1.1 or lower : https://sites.google.com/site/appleclubfhs/support/advice-and-articles/finding-a pps-for-older-devices#5.1.1