Thanks to a lot of work by Daniele Cordella and just a very little bit from me, the Contact Form block has been updated for Moodle 1.9.x and is available from the contrib 19STABLE branch, or by clicking the 'Download latest version' in its entry in the Module and Plugins database.
(And a huge thank you to Anthony Borrow for helping make all this happen correctly!)
Please note that this version is only for Moodle 1.9.x. It will not work on previous versions of Moodle and it has not been tested on Moodle 2.0 dev.
This version uses the Moodle forms library and supports recaptcha if you have enabled it on your site. It also uses roles to control who receives the emails submitted, so please read the README.TXT file, and feel free to ask if you have any questions or problems.
Daniele and I both have this on production Moodle installs without any problems, but please report any bugs or errors that you get when using the block.