You can send a copy of the file to the PDF Expert app from other apps using the Open in dialogue (please note: only files can be exported to PDF Expert; notes or web pages cannot be shared to PDF Expert, they need to be converted to files first).
If the PDF Expert is not in the list of available apps, please make sure the latest iOS version is in use on your device. If updating iOS does not help, scroll the list of apps until you see the More button, tap it, then drag the PDF Expert app thumbnail to the top of the list, disable toggles of apps you do not need in the Open in dialogue, restart the device and check the situation after that.
You can also save files to a specific folder in the PDF Expert 6 app (e.g. when saving the PDF file from Safari) with the Files app:
Note: if PDF Expert is not in the list of locations in the Files app, open the Files app > 'Browse' view, tap 'Edit' in the top right corner, enable PDF Expert toggle in the list of Locations, tap 'Done'. Now you will have access to the local files and folders in PDF Expert 6 via the Files app.