After looking at the Apple Business Manager site, I see that my information above is incorrect.
You do want the Apple Business Manager: https://developer.apple.com/business/custom-apps/
This page has some details about private distribution: https://developer.apple.com/business/distribute/
The footnote on that last page mentions the difference between Apple Business Manager and the Apple Enterprise Developer Program. The Apple Enterprise program is truly private. You are only allowed to distribute apps to your own employees. Those apps do not have to pass Apple's app review. Some developers have been abusing this program and publicly distributing apps.
Unfortunately, the Apple Business Manager is not very straightforward. When I try to learn more about it, some pages seem to talk about it from the developer's perspective. Other pages seem to talk about it from the client's perspective. This help page seems to have the best information: https://help.apple.com/app-store-connect/#/dev275598f16
It includes a link to the Getting Started Guide.
What you have to is create a new app. Before it has been approved, you can specify individual organizations that can access it.