you don't need SP Server on-premises, but you will need some 3rd party tools to migrate some of the content. Depending on what the Notes apps do, you will need to rebuild/replace them with some different technologies
I know that Quest has a tool, but I've never used it so I have no idea how well it works.
This may be a good opportunity to reimagine the applications by reviewing the business processes to see what is going to be needed in the future. It is quite possible that the old apps are no longer what the business users really need.
Agree with Dean you have to look for third party tools to do the migration or even built your own migration stuff...one important thing to consider here is about what you need to migrate...if we are talking about data, what has been explained applies...but if we are talking about a full Notes App I'm afraid you will have to rewrite it in SPO / Office 365 / Azure depending on what your App does
@James_DanielI would recommend you to contact us at Binary Tree (www.binarytree.com), we are helping many customers to migrate big volumes of Domino databases to Sharepoint Online. You can also send me an email at Viktor.firstname.lastname@example.org if you want to know more of what we do :)