Can a SaaS provider terminate a customer's access to their software without notice?

A SaaS agreement is just like every other business contract with just a few notable exceptions. They can be "click-wrap" agreements where your users are not able to negotiate the terms of service. These agreements must be inherently reasonable and fair aligned to traditional notions of fair play. If you intend for a more enterprise service, then your agreements can be a more traditional bargain for exchange so long as they are not deceptive and misleading. The other major difference with SaaS agreements is typically their ability to be unilaterally updated when written properly and applied fairly. You should definitely work with a lawyer experienced with data security and privacy issues to write your agreement and ensure it aligns to your terms and services, privacy policy, and internal business practices.

These are questions that you should discuss with a Lawyer. Without seeing the content of the contract and understanding the nature of your business, I doubt anyone will be able to provide you actionable advice.

To be able to answer your question, I would need to see the SaaS agreement you have with your potential customer. I could then let you know your rights, and what you need to say in the Agreement to be fully protected and retain your data.

To help you, an attorney would have to review the agreement. I suggest you contact a business attorney to help you with this. You can contact me through this website, and I will be happy to walk you through this.

What you need and ask for are extensive, and cannot be given on a question and answer basis on a site like this. I suggest you engage a business attorney to help you with these endeavors.