Hello Ch3shireCat,
When you package out a game, it will come with the required pre-requisites in the following location:
WindowsNoEditor\Engine\Extras\Redist\en-us
Installing the pre-requisites listed there should be able to avoid any issues like this. Keep in mind that it could still cause errors like the one you showed as that can sometimes be the symptom of a corrupt installation, where it will need to be uninstalled and reinstalled as attempting to repair it rarely works.
These prerequisites will be required in any case, whether they are already installed or newly installed with your game.