Its source is from LRCLIB (lrclib.net). LRCLIB collects lyrics from various sources, including MXM, but also allows users to submit their own synced lyrics.
You can understand it as a "Library Genesis" for lyrics. LRCLIB will also become open-source (soon™).
DMCA is not available in every countries. About applications that are published to Appstore/Play Store, they may ask for users to manually enter a 'LRCAPI-compatible instance' URL to avoid issues.
I'm not a foobar2000 user, but after some searches, there are Lyrics Show Panel 3 (mostly dead?) and its opensource alternative foo_openlyrics (https://github.com/jacquesh/foo_openlyrics).
Adding LRCLIB source should be easy and straightforward enough (https://github.com/jacquesh/foo_openlyrics/tree/master/src/s...). I'm not very experienced in C++ though, so I suggest opening an issue first to see if the author or anyone are willing to do the intergration.