Dynamic SQL With Double Quoting
You can temporarily change QUOTED_IDENTIFIER.

DECLARE @TestHyphens TABLE(StoreName varchar(500),topItemSale varchar(500))

INSERT INTO @TestHyphens (StoreName, topItemSale) VALUES
('Bob''s Burger''s','Pen'),
('Jacks Coffee Shop','Pencil'),
('Larry''s Hair Shack','Rainbow'), 
('Manny''s Sandwich Shop','Sunglasses');

SET QUOTED_IDENTIFIER OFF

SELECT StoreName, TopItemSale 
FROM   @TestHyphens 
WHERE  StoreName = "Bob's Burger's"

SET QUOTED_IDENTIFIER ON

| StoreName      | topItemSale |
|----------------|-------------|
| Bob's Burger's | Pen         |


Rextester here

Using within the cursor

DECLARE @sql nvarchar(MAX), @storename varchar(500)

DECLARE db_cursor CURSOR FOR
SELECT StoreName
FROM #TestHyphens

OPEN db_cursor
FETCH NEXT FROM db_cursor INTO @storename

WHILE @@FETCH_STATUS = 0
BEGIN

    SET @SQL = N'SET QUOTED_IDENTIFIER OFF '
               + N'Select * from #TestHyphens where StoreName = "' + @storename + N'"'
               + N'SET QUOTED_IDENTIFIER ON';

    EXECUTE SP_EXECUTESQL @SQL;    

    FETCH NEXT FROM db_cursor INTO @storename
END

CLOSE db_cursor
DEALLOCATE db_cursor


This is the result:

| StoreName             | topItemSale |
|-----------------------|-------------|
| Bob's Burger's        | Pen         |
| Jacks Coffee Shop     | Pencil      |
| Larry's Hair Shack    | Rainbow     |
| Manny's Sandwich Shop | Sunglasses  |


Rextester here