Ok, another quick update. I found out the vBulletin code to get the logouthash:

$user['securitytoken_raw'] = sha1($user['userid'] . sha1($user['salt']) . sha1(COOKIE_SALT));
$user['securitytoken'] = TIMENOW . '-' . sha1(TIMENOW . $user['securitytoken_raw']);
$user['logouthash'] =& $user['securitytoken'];

TIMENOW is supposed to be the current time in UNIX time, so I got to the following:

my $logouthash = time . '-' . sha1_hex(time . $vb_id . sha1_hex($usersalt) . sha1_hex('My VB license number'));

So now the logouthash looks like it shows up in correct format, but it still doesn't work. Maybe the current time in UNIX time in perl is not "time"? I'm gonna look further into what else I might've done wrong. I think I'm getting closer to a solution... :D