How does copyleft license work in this case? Aren't they legally required to open-source their sources and publish it publicly or can they hide that behind a request system?
Unlike the full GPL, the LGPL doesn't require applications of the library be licensed under the same. It would apply only to improvements to the library itself.
No, the main requirement is that they link to libfprint dynamically or provide object files such that effectively you can replace libfprint with a different version of libfprint of your choosing, but there's no requirement to open source the work that makes use of libfprint.