Running Total to the latest period

OK, you haven't had an answer in a full day so even though I work in TSQL I'll try a solution that's ANSI SQL compatible. Work with me if my syntax is off a bit.

Before we start, check your "Desired Output", you're currently showing a running total for A on 3/22 of 5, but you had a 2 for A on 2/22 so it should be a running total of 7, right?

Anyway, assuming that's just a typo, I'd approach this by making a few CTEs that build a list of all {PERIOD, NAME} pairs you want reported, then JOIN your actual data to that. There are a number of ways to generate the dates, the easiest is to use DISTINCT if your actual data is fairly robust, but I can describe other methods if that assumption does not hold for your data.

So with all that in mind, here is my solution. I put your sample data in a CTE for portability, just replace my "cteTabA" with whatever your data table is really named

--Code sample data as a CTE for portability
;with cteTabA as ( 
    SELECT * 
    FROM ( VALUES 
        ('A', '02/2022', '2')
        , ('A', '03/2022', '5')
        , ('A', '04/2022', '3')
        , ('A', '05/2022', '7')
        , ('B', '02/2022', '9')
        , ('B', '04/2022', '6')
    ) as TabA(Name, Period, Value) 
) --END of sample data, actual query below
--First, build a list of periods to use. If your data set is full, just select DISTINCT
, cteDates as ( --but there are other ways if this doesn't work for you - let me know!
    SELECT DISTINCT Period FROM cteTabA
) --Next, build a list of names to report on
, cteNames as (
    SELECT DISTINCT Name FROM cteTabA
) --Now build your table that has all periods for all names
, cteRepOn as (
    SELECT * FROM cteNames CROSS JOIN cteDates 
)--Now assemble a table that has entries for each period for each name,
--but fill in zeroes for those you don't actually have data for
, cteFullList as (
    SELECT L.*, COALESCE(D.Value, 0) as Value
    FROM cteRepOn as L 
        LEFT OUTER JOIN cteTabA as D on L.Name = D.Name AND L.Period = D.Period 
)--Now your query works as expected with the gaps filled in
SELECT PERIOD, NAME, Value 
    ,SUM(Value) OVER (PARTITION BY NAME ORDER BY PERIOD) AS balance
FROM  cteFullList
WHERE Period < '06/2022'--CURRENT_DATE()
ORDER BY NAME, PERIOD

This produces an output as follows