First, reference this article and ensure that all the necessary servers are accessible via your network and computer.
If you've determined that all the servers are accessible, you should then troubleshoot how LockDown Browser was installed on the Mac(s). Using a silent install process via a Mac installer is not supported, and has been known to result in the A19 error. You should only use the installation process described in this article.
As a test, download the student edition of LockDown Browser and install it on one of the effected computers. If the A19 error does not occur afterwards, then it's likely that the original installation process that was used was incorrect.