Preferably via a mailing list, which you can subscribe by sending empty e-mail with subject "subscribe lms-en" to address firstname.lastname@example.org. Further emails should be sent to address email@example.com.
In order to effectively report bugs or new ideas, it's recommended to subscribe mailing list, where you can get quickest response for your problems from either LMS users or developers team. There's also Bug Tracking System available, when you can report bugs upon registration. Reports from BTS are also forwarded to mailing list, so the best practice is to subscribe to list, report via BTS and send link to mailing list and finally wait as your report will evolve. BTS address is bts.lms.org.pl.
Newest version of LMS can be downloaded from CVS repository utilizing WWW interface at here, or with CVS tools (anonymous access, empty password):
cvs -d :pserver:firstname.lastname@example.org:/cvsroot login cvs -d :pserver:email@example.com:/cvsroot co lms cvs -d :pserver:firstname.lastname@example.org:/cvsroot logout
Information about changes and version history of LMS is included in Changelog.
There are questions appearing periodically on lms mailing lists about adding some funtions to LMS. Matter of payment for support is very difficult. Note, that there is no company behind LMS, so any commercial adoption or support should be discussed directly between You and involved developer. You are reading english version of documentation, it's likely that you are not from Poland, which makes even simple things like issuing invoice or money transfer complex. The most experienced developer in that matter is Alec. You can see details on support on his page: http://lms.alec.pl/index.php?lang=en