Generally, a role with just 21 capabilities: read, view_admin_dashboard and all 18 caps from the “Custom Post Types->Coupons” group provides access for working with WooCommerce coupons and adding a new coupon:
add new coupon
coupons access list

Try to deactivate all plugins and re-test. If some plugin prevents creating a new coupon, activate plugins back one by one to isolate a conflict.