Order placement on the Drupal website

The checkout process /cart  then /cart/checkout works fine in all browsers, but when clicking the continue button in Firefox and Chrome, everything is great - the order completes, while in IE and Opera nothing happens (the button is active - not overridden by styles). I've been digging for several days now, but haven't found anything yet. Have you encountered this issue in your experience?
I would greatly appreciate any response.
p.s. Thank you for your project; I have been using it for a long time, but only today did I decide to register.