You can do a one-time update using the below query
UPDATE t1
SET t1.[Company_Name_Lookup] = t2.[Company_Name]
FROM dbo.audit AS t1
INNER JOIN dbo.company AS t2
ON t1.[Company_ID] = t2.[Company_ID]
WHERE t1.[Company_Name_Lookup] is NULL;
For subsequent records in audit table, you can write a trigger with similiar logic as above to populate the Company_Name_Lookup column.
However, this is not very efficient because you are duplicating the Company_name in your audit table even when you have a Company_ID column.
Where do you plan to use the Audit table?
How big (number of records) is this table expected to grow?
You will need to provide more details about your use case so that we can provide better suggestions.