Yes, this is fine with a single personal license – as long as it’s only you who is operating those workstations and only you have access to the Prodigy app and Python library
By default, it uses a local SQLite database, but you can also plug in a remote MySQL or PostgreSQL database, or your own custom implementation.
Under the hood, we use peewee, so everything that works with peewee will work pretty much out-of-the-box. The PRODIGY_README.html that's included with Prodigy also contains the detailed API docs of the internal Database class and its methods, so you could also implement and add something entirely custom and use a different database.