Do purchased Apple Books require Wi-Fi

Only if you haven't downloaded it to the iPad.

If you download it while connected to the internet, it will remain available when you are offline.

When you open a purchased book, it should automatically download to the iPad.

If it has a cloud icon, it has not been downloaded yet.

As correctly mentioned, Apple Books operates fully offline, once the purchased book has been downloaded.

PDF files and EPUB books acquired elsewhere are also fully accessible offline, once downloaded form iCloud.

If you have enabled the optimized storage management feature, you can selectively override it and keep specific books downloaded, as well: Transfer files from iPad to a storage device, a server, or the cloud - Apple Support