To process a VIVO mobile order, enter the order ID in the Express Checkout filter to access the order page. Fill in details like invoice number, IMEI number, and customer details. Enter the pickup code provided by the customer within 24 hours and click "Handover Package" to mark the order as delivered. It is important to follow all processing steps correctly and not hand over the mobile before completing the order to ensure the customer receives offers and the seller receives payout.