The Quick Answer
Wholesale Now: No, shoppers cannot use discount codes at checkout.
Explanation
Wholesale Now uses Shopify's 'draft orders' feature. Unfortunately, draft orders do not support discount codes on the checkout page.
This is a limitation in Shopify - not our apps (https://community.shopify.com/c/shopify-discussions/draft-orders-no-codes-allowed/m-p/1257884)
Note: Old, not working link: https://ecommerce.shopify.com/c/shopify-discussion/t/draft-orders-in-admin-is-great-but-why-is-it-missing-discount-codes-324056
However, Bulk Discount 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.
We plan to bring this feature over to Wholesale Now, however, we don't have a timeline yet for this.