Easiest solution may be to delete the app and install it again. Then log in to your Ring account. This should prompt you to allow notifications.
You also may want to download the RapidRing app (it is an official Ring app) which is super fast in showing you live video.
You want to check settings in both the Ring app and iOS settings.
On the Ring App, click on the 3 lines (top left icon), then Devices, and select your Doorbell. Here first make sure Ring Alerts is turned on (if you want the doorbell ringing to alert you) and Motion Alerts turned on (if you want alerts if someone is in front of it). Then go to Device Settings, and then Alert Settings and finally Notification Preferences and verify In-App Notifications is turned on.
On iOS, go to Settings, Notifications, Ring, and make sure Allow Notifcations is turned ON.
If you use the RapidRing app as well, by default you will get notifications from both apps. In this case, you may want to turn off Ring and Motion notifications from the Ring app, but keep them on from RapidRing.