Hi,
I was able to access the website and check on the problem.
From what I can see, the problem is that you customized the file "payment" of the view "checkout" via the menu Display>Views and you somehow don't have anymore this code:
if($this->config->get('auto_submit_methods',1) && empty($method->ask_cc) && empty($method->custom_html) && empty($checked)){
$checked.=' onclick="this.form.action=this.form.action+\'#hikashop_payment_methods\';this.form.submit(); return false;"';
}
which handles the auto submit of the payment method and thus avoids the need to click a second time on the finish button.