If I buy an iPhone 5s, will I still be able to go on the appstore and download apps? Just don't know if the appstore will allow phones that aren't running the current OS to go on it

(I'm really only interested in this app - https://www.literatureandlatte.com/scrivener/overview )
Yes, the App Store is certainly supported on the iPhone 5S.

Note that the iPhone 5S does not supported the just released iOS 13 release. This means that apps that require iOS 13 or later won't be installable on the iPhone 5S. For the moment, only very few apps require iOS 13.

If an app does require iOS 13, and is not available for download, usually the publisher will let older versions of the app that does not require iOS 13 to be available for download on those older devices.

The app you mention (Scrivener) only requires iOS 9 - so it's not going to be a problem for you to download and run this app on the iPhone 5S. In the future it might be that you cannot update to the latest version of the app, if they release a version that requires iOS 13 though.