A "company name" field for either the bill to address or ship to address is not a standard field with Yahoo!Store. If you setup your custom field within Yahoo! appropriately there is no need for any changes to be made with in the Order Manager.
From the Yahoo! Store Manager, select Order Form.
Find the section marked "Billing Info" and select Edit Extra Fields.
Click Add New Field and fill in the form seen below.
You MUST set the value for "name" to either "Company" or "Company Name"; anything else will not work. Your customers will see whatever you enter in the "Text" field.
Click Update.
From the Yahoo! Store Manager, select Order Form.
Find the section marked "Shipping Info" and select Edit Extra Fields.
Repeat steps three to five.