iTunes songs do not contain lyrics unless you have added them yourself using the computer version of iTunes. You do this by right clicking on the track, choosing Get Info, and pasting the lyrics into the Lyrics tab. I seem to remember hearing about some computer side tools that can automate this with some success, but am not sure.
Once added to iTunes on the computer they should sync to the iPad the next time you sync your music with that computer/iTunes.
If there is any other way to add lyrics to iTunes purchased songs on the iPad, I am not aware of it.
The last time I checked the iPad's Music app did not display lyrics, though the iPhone version does. That was a few updates ago, os YMMV.
There may be a way of adding songs/lyrics directly to AnyTunes. I'm not familiar with the app. If you have not done so I'd go to the developer's website (link fund on their App Store page). If the answer isn't there already, there is usually a way to ask them for help.
Good luck.