Performing an UPDATE with Union in SQL

You do not need a UNION for that - replacing an inner join with a pair of outer ones should do it:

UPDATE c
SET c.order_status = 1
FROM Customer AS c
LEFT OUTER JOIN Order_Web As ow ON c.id = ow.customer_id
LEFT OUTER JOIN Order As o ON c.id = o.customer_id
WHERE ow.order_filled = 1 OR o.order_filled = 1

You could also use a WHERE EXISTS, like this:

UPDATE c
SET c.order_status = 1
FROM Customer AS c
WHERE EXISTS (
    SELECT 1 FROM Order_Web As ow WHERE c.id = ow.customer_id AND ow.order_filled = 1 
) OR EXISTS (
    SELECT 1 FROM Order As o WHERE c.id = o.customer_id AND o.order_filled = 1
)

If you must use UNION, you can do it as follows:

UPDATE c
SET c.order_status = 1
FROM Customer AS c
WHERE c.id in (
        SELECT ow.id FROM Order_Web As ow WHERE ow.order_filled = 1 
    UNION
        SELECT o.id FROM Order As o WHERE o.order_filled = 1
)

or the same one with a JOIN:

UPDATE c
SET c.order_status = 1
FROM Customer AS c
JOIN (
        SELECT ow.id AS id FROM Order_Web As ow WHERE ow.order_filled = 1 
    UNION
        SELECT o.id AS id FROM Order As o WHERE o.order_filled = 1
) AS ids ON ids.id = c.id

UPDATE  c
SET     c.order_status = 1
FROM    (
        SELECT  customer_id
        FROM    order_web
        WHERE   order_filled = 1
        UNION
        SELECT  customer_id
        FROM    order
        WHERE   order_filled = 1
        ) o
JOIN    customer c
ON      c.id = o.customer_id