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 app on iOS 7 (with the main app, but not the widget working) or do I have to create an iOS 7 version without the widget?
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.