I’m Poppy one of the Guru Support team at Shopify.
I want to make sure I'm answering your question properly, so I wanted to double check are you referring to the order notes option on the Cart page? If so, there are a few different possible solutions for making the customer order notes required.
The first, would be to use an app. I found a couple in the app store that may work depending on what you're using the custom notes for:
The second option would be to code in the form, instead of using the option in the theme editor, and make the input required. This option is explained in a bit more detail on this forum post. The post also includes a link to a code generator specifically for Shopify Forms. I always recommend our merchants duplicate their themes before trying out any new code. You can roll back some of the theme files, but its good to have a back up just in case.
If you don't feel comfortable changing the code, or the apps don't feel like the right fight, you could always reach out to a Shopify Expert. You can discuss the options with them for having something like this custom coded into your store.
If I've misunderstood your question please let me know and I'll take another look! You can give us a call too or start a live chat at any time. We're here 24/7 and are always happy to help out!
Thank you, :)