One possible way of addressing this problem is to:

• select all possible accounts from your table
• generate a calendar table in range (minimum_date, maximum_date) belonging to your table dates
• cross join all possible accounts with all possible dates, then left join this to your table, and fire your running sum

WITH cte_accounts AS(
    SELECT DISTINCT Account FROM tab
), cte_dates AS (
    SELECT MIN(Date) AS date_val, MAX(Date) AS last_date_val FROM tab
    UNION ALL 
    SELECT DATEADD(DAY, 1, date_val), last_date_VAL FROM cte_dates WHERE last_date_val > date_val 
)
SELECT a.Account,  
       d.date_val,  
       SUM(tab.Amount) OVER (PARTITION BY a.Account ORDER BY d.date_val) AS RunningTotal
FROM       cte_accounts a
CROSS JOIN cte_dates    d
LEFT JOIN tab 
       ON tab.Account = a.Account
      AND tab.date = d.date_val
ORDER BY d.date_val, a.Account

Output

Check the demo here .

The last ORDER BY clause is not necessary: it's just for visualization purposes. On a side note, if you do mind about having the null values, you can filter them out in a subquery.

Another example, similar to @lemon answer. Added checking for first date for Account. Without full calendar.

with AccDates as(
  select  Account, min(date) minDate
  from tab1
  group by account
)
,Dates as( select distinct date from tab1)

select a.Account
  ,d.date
  ,sum(t.Amount)over(partition by a.Account order by d.Date) RunningTotal
from AccDates a 
left join Dates d on  d.date>=a.minDate
left join tab1 t on t.Account=a.Account and t.date=d.date
order by d.date,a.account
;

Result