Database Migration from legacy app on old iPad
Unfortunately it is completely irresponsible to store patient records without regards for security, backups and vendor support, and without being aware of the fact that the vendor discontinued the product and removed it from the App Store years ago. In many countries you would risk being fined heavily by the authorities.
At this point, you have a few options:

Contact the vendor offering money to have them help you migrate the data.

Contract a third party developer or IT-professional that can help you migrate the data (you could be 'lucky' that the data is simply stored in a SQLite database on device)

Migrate the data manually


I would strongly suggest initiating option 3 immediately while simualtenously pursuing the first two options.
Migrating the data manually could be done by simply going through all data in the app and taking screenshots along the way. Make sure to handle those screenshots in a professional and ethical manner - i.e. ensure they're not synced with Dropbox or Google Photos or something like that - and transfer them to a secure PC, where you can transfer the data into a new patient records system.