Hello rosieeiphone,
Thanks for that info and for choosing the Apple Support Communities. If we understand correctly, you are unable to see the Reminders app on your iPhone, but if you search it on the App Store app it shows an Open option. This sounds like the app is still installed on your iPhone but just not in a place that you can find it, so please see these Apple resources on how to find and put the Reminders back to a place you can find it.
Organize the Home Screen and App Library on your iPhone
How to move apps and create folders on your iPhone, iPad, or iPod touch
If you are still having an issue with finding the Reminders app, please go to Settings > General > Reset and tap the "Reset Home Screen Layout" option and then go back to your Home Screen and check for the Reminders app again.
Return iPhone settings to their defaults
Cheers!