I have searched the forum many times but couldn't find a solution for my situation. I am working with an Oracle database.

I have a table with all Order Numbers and Customer Numbers by Day. It looks like this:

Day | Customer Nbr | Order Nbr 
2018-01-05 | 25687459 | 256 
2018-01-09 | 36478592 | 398 
2018-03-07 | 25687459 | 1547 
and so on....

Now I need a SQL Query which gives me a table by day and Customer Nbr and counts the number of unique Order Numbers within the last 365 days starting from column 1.

For the example above the resulting table should look like:

Day | Customer Nbr | Order Cnt 
2019-01-01 | 25687459 | 2 
2019-01-02 | 25687459 | 2 
... 
2019-03-01 | 25687459 | 1 

One method is to generate values for all days of interest for each customer and then use a correlated subquery:

with dates as (
 select date '2019-01-01' + rownum as dte from dual
 connect by date '2019-01-01' + rownum < sysdate
 )
select d.dte, t.customer_nbr,
 (select count(*)
 from t t2
 where t2.customer_nbr = t.customer_nbr and
 t2.day <= t.dte and
 t2.date > t.dte - 365
 ) as order_cnt
from dates d cross join
 (select distinct customer_nbr from t) ;

Edit:

I've just seen you clarify the question, which I've interpreted to mean:
For every day in the last year, show how many orders there were for each customer between that date, and 1 year previously. Working on an answer now...

Updated Answer:

For each customer, we count the number of records between the order day, and 365 days before it...

WITH yourTable AS
(
 SELECT SYSDATE - 1 Day, 'Alex' CustomerNbr FROM DUAL
 UNION ALL
 SELECT SYSDATE - 2, 'Alex' FROM DUAL
 UNION ALL
 SELECT SYSDATE - 366, 'Alex'FROM DUAL

 UNION ALL
 SELECT SYSDATE - 400, 'Alex'FROM DUAL
 UNION ALL
 SELECT SYSDATE - 500, 'Alex'FROM DUAL
 UNION ALL
 SELECT SYSDATE - 1, 'Joe'FROM DUAL
 UNION ALL
 SELECT SYSDATE - 300, 'Chris'FROM DUAL
 UNION ALL
 SELECT SYSDATE - 1, 'Chris'FROM DUAL
)

SELECT Day, CustomerNbr, OrdersLast365Days
FROM yourTable t
OUTER APPLY
(
 SELECT COUNT(1) OrdersLast365Days
 FROM yourTable t2
 WHERE t.CustomerNbr = t2.CustomerNbr
 AND TRUNC(t2.Day) >= TRUNC(t.Day) - 364
 AND TRUNC(t2.Day) <= TRUNC(t.Day)
)
ORDER BY t.Day DESC, t.CustomerNbr;

If you want to report on just the days you have orders for, then a simple WHERE clause should be enough:

SELECT Day, CustomerNbr, COUNT(1) OrderCount
FROM <yourTable>
WHERE TRUNC(DAY) >= TRUNC(SYSDATE -364)
GROUP BY Day, CustomerNbr
ORDER BY Day Desc;

If you want to report on every day, you'll need to generate them first. This can be done by a recursive CTE, which you then join to your table:

WITH last365Days AS
(
 SELECT TRUNC (SYSDATE - ROWNUM + 1) dt
 FROM DUAL CONNECT BY ROWNUM < 365
)

SELECT d.Day, COALESCE(t.CustomerNbr, 'None') CustomerNbr, SUM(CASE WHEN t.CustomerNbr IS NULL THEN 0 ELSE 1 END) OrderCount
FROM last365Days d
LEFT OUTER JOIN <yourTable> t
 ON d.Day = TRUNC(t.Day)
GROUP BY d.Day, t.CustomerNbr
ORDER BY d.Day Desc;

I would probably have done it with and analytic function. In your windowing clause, you can specify a number of rows before, or a range. In this case I will use a range.

This will give you, For Each customer for each day the number of orders during one rolling year before the date displayed

WITH DATES AS ( 
 SELECT * FROM 
 (SELECT TRUNC(SYSDATE)-(LEVEL-1) AS DAY FROM DUAL CONNECT BY TRUNC(SYSDATE)-(LEVEL-1) >= ( SELECT MIN(TRUNC(DAY)) FROM MY_TABLE )) 
 CROSS JOIN 
 (SELECT DISTINCT CUST_ID FROM MY_TABLE))
SELECT DISTINCT
 DATES.DAY,
 DATES.CUST_ID,
 COUNT(ORDER_ID) OVER (PARTITION BY DATES.CUST_ID ORDER BY DATES.DAY RANGE BETWEEN INTERVAL '1' YEAR PRECEDING AND INTERVAL '1' SECOND PRECEDING) 
FROM
 DATES
LEFT JOIN 
 MY_TABLE 
ON DATES.DAY=TRUNC(MY_TABLE.DAY) AND DATES.CUST_ID=MY_TABLE.CUST_ID
ORDER BY DATES.CUST_ID,DATES.DAY;