The coupon field will only show up on the core checkout page if there are available coupons


The Checkout Core Page is a dynamic page. The fields displayed change based on the payment settings configured, product settings configuration, whether the person viewing the page is logged in or not, as well as the embedded information provided from purchase links


If you have available coupons, then you should double-check that the [MM_Form_Section type='coupon'] and [/MM_Form_Section] are surrounding the coupon code on your checkout page.