The issue appears to be that the template has been overwritten with some custom code.
The source code should have a form called “pwgc-redeem-gift-card-form”, however when I look at your HTML it is missing.
The “Apply Gift Card” box is currently located above the “Proceed to checkout” button (below the totals). We have had another user request the ability to move it up to beneath the “Apply Coupon” button, however that isn’t currently an option yet.
If you want to be notified when we make that an option I can contact you via this thread. It will be a couple of weeks but it is something we have planned.
In the meantime, you will need to remove the custom template file for /woocommerce/cart/apply-gift-card.php, or the changes made to functions.php since that is what is interfering with the Apply Gift Card form.