The Quick Answer
Wholesale Now: No, shoppers cannot use discount codes at checkout.
Quantity Breaks Now: Shoppers can use discount codes on the cart page but not at checkout (see below for details.)
Wholesale Now and Quantity Breaks Now use Shopify's 'draft orders' feature. Unfortunately draft orders do not support discount codes in the checkout page.
This is a limitation in Shopify - not our apps (https://ecommerce.shopify.com/c/shopify-discussion/t/draft-orders-in-admin-is-great-but-why-is-it-missing-discount-codes-324056)
However, Quantity Breaks Now does have a feature that allows your shoppers to use discount codes while on the cart page -- as a workaround to the draft order checkout page not having the discount code field.
This is found under the Settings tab in the Cart Page section:
We plan to bring this feature over to Wholesale Now, however we don't have a timeline yet for this.