I think you will need to join on the stop table twice once for the stop_type of PU and once for the stop_type of SO. Something like below:

SELECT o.rate,
sPU.actual_departure,
sPU.city_name,
sPU.state,
o.id,
PickUpCIty = sPU.City_Name,
Delivery =sSO.city_name
FROM orders o
RIGHT OUTER JOIN stop sPU ON o.id = sPU.order_id AND sPU.Stop_Type = 'PU' -- Pickup CIty 
RIGHT OUTER JOIN stop sSO ON o.id = sSO.order_id AND sSO.Stop_Type = 'SO' -- Delivery City
WHERE o.equipment_type_id = 'V'
AND o.customer_id = 'BEACCOOH'
AND o.status = 'D'; 

You may need to make one or both of the joins to stop left joins.