Is There a Permission in the Employee Roles That Makes Invoices Delivered or Received Automatically?

There is no separate permission in the employee roles that makes invoices delivered or received automatically. To set this for new invoices, use the following settings:

• Sales invoices: Enable Mark Sales Invoices as Delivered by Default from Invoicing Settings in the section Inventory Control, so the box Already Delivered is selected in new sales invoices. This setting applies only when the requisitions for sales invoices are disabled.
• Purchase invoices: Enable Mark Purchase Invoices as Received by Default from Purchase Invoice Settings. This setting applies when the requisitions for purchase invoices are disabled.