A family of Microsoft relational database management systems designed for ease of use.
How are you running the Update query? Note that opening it in datasheet view does NOT actually execute it - instead it shows the data prior to its execution. The ! icon on the toolbar will run the query (you should get a message box saying "this query will make permanent changes...")
I'd suggest as SQL:
UPDATE CSCust
SET CSCust.final_company = Replace(Replace([final_company], Chr(34), ""),Chr(39), "")
WHERE final_company LIKE "*" & Chr(34) & "*"
OR final_company LIKE "*" & Chr(39) & "*";
to limit the update to those records which need updating.