There are lot of ways of doing this:

Old skool:

SELECT  id_num
,   CASE WHEN provider = '' THEN (SELECT    TOP 1 Provider FROM #TEST_INSURANCE_PAYMENTS tt WHERE   tt.id_num < t.id_num AND Provider <> '' ORDER BY id_num DESC) ELSE provider END
,   location, last_update, Total_Charge
FROM    #TEST_INSURANCE_PAYMENTS t

This fetches the last non-empty provider from previous rows if current one is empty. Downside is extra join.

New hotness:

SELECT  id_num
,   CASE 
        WHEN provider = '' THEN STUFF(MAX(RIGHT(CONCAT('0000000000',id_num), 10) + CASE WHEN Provider <> '' THEN Provider END) OVER(ORDER BY id_num), 1, 10, '')
        ELSE provider END
,   location, last_update, Total_Charge
FROM    #TEST_INSURANCE_PAYMENTS t

I don't know the name of this technique but what it does is creates a single frame of combination of id_num + provider, like: 0000000001Vimalkumar Veerappan. Where provider is empty, we change it to NULL, so value becomes NULL.

By taking MAX(...) OVER (ORDER BY id_num) call, we fetch the last value in that frame. Then, by doing STUFF(...) we remove the 0000000001 part, and what's left is what we need.

This technique enables a single windowed call, which might generate best performance, downside is somewhat obscure code.

We won't be able to tell what's going on without a plan (as Dale suggests). In the mean time, maybe this will be a little better:

SELECT x.id_num, COALESCE(NULLIF(x.Provider,' '),y.provider) AS [Provider], x.Location, x.last_update, x.Total_Charge
  FROM @TEST_INSURANCE_PAYMENTS x
    OUTER APPLY (SELECT MAX(id_num) AS mid_num FROM @TEST_INSURANCE_PAYMENTS y WHERE y.id_num < x.id_num AND y.[Provider] IS NOT NULL AND y.Provider <> ' ' AND x.Provider = '') a
    LEFT OUTER JOIN @TEST_INSURANCE_PAYMENTS y
      ON a.mid_num = y.id_num;

For Azure or SQL Server 2022, you can just use LAG() with IGNORE NULLS

This avoids JOINS correlated subqueries, etc, etc, and only scans the table once.

SELECT
  *,
  LAG(
    NULLIF(Provider, ' '),
    0
  )
    IGNORE NULLS
    OVER (
      ORDER BY id_num
    )
FROM
  #TEST_INSURANCE_PAYMENTS

fiddle