In my case I double checked my Discounts in Shopify Admin.
It turned out that I did not have any active coupons. All of my coupons were set to be active in the future.
So as part of your troubleshooting, double check the active dates (including expiration date) on your coupons. Usage limits, etc.
Because it appears that Shopify will only show the Coupon Code feature on checkout if your site has active coupons and no other restrictions set in the coupon code admin page.
Rowby
Shopify Partner
Los Angeles