Hi sd1cko,
Thanks for your post. I can fully understand that you want to migrate email accounts and the mailbox items to Office 365.
Currently, there is no Microsoft supported article to migrate from SBS 2003 to Office 365 (Exchange Online). The article mentioned above applies to the standard Exchange server 2003 or later version, but not SBS. However, based on my experience, the steps to migrate from SBS 2003 to Office 365 are very identical to the steps for standard Exchange 2010.
Anyway, you can also consider the PST migration. Here are the general steps for your reference:
1. Export the emails from the old email account to a PST file in Outlook client.
2. Sign up for the Office 365 subscription and create the Office 365 accounts, and then automatically configure the Office 365 account in Outlook client via Autodiscover service.
3. Import the .PST file to the Office 365 account.
Moreover, here is a related article for your reference: Use Outlook to Move Data Between Accounts.
For PST migration, that all the users need to perform the PST migration respectively.
After all the data is migrated to Office 365, you can add the custom domain (companyname.com) in your Office 365 tenant and configure the required DNS record for Office 365, and change the domain suffix from ourcompany.onmicrosoft.com to companyname.com.
If anything is not so clear, we are glad to provide further assistance.
Best regards,
Young Yang