I am creating an iOS app in swift that has a Notification Center widget.
However, I want the app to run on iOS 7 just fine. If I have the widget, can I just run the
Yes, you can. Quoting Apple documentation:
If you link to an embedded framework from your containing app, you can still deploy it to versions of iOS older than 8.0, even though embedded frameworks are not available in those versions.
And this was in the App Extension Programming Guide. Therefore, it's only axiomatic that extensions can be deployed with apps to older versions of iOS.
Yes, it will work on iOS 7 just fine (if your deployment target is iOS 7 or lower).