You may not want an MDM, but the only way to distribute apps through the volume purchase program is to have an MDM.
You must understand the difference between ABM and an MDM. ABM is a binding association between your business and Apple and it establishes chain of custody for hard (Macs, iPads, iPhones, etc.) and soft (App Store apps, books) assets. ABM is about establishing and enforcing custody. ABM is not about managing those hard or soft assets. Management is provided by an MDM. Proof of ownership/control is provided by ABM.
Next, you have entered into the second realm of ABM... identity. By creating Managed Apple IDs (MAIDs), you are establishing a chain of custody to an identity. This is true if you did this the right way but federation and sync, or the hard way by manually creating MAIDs. In either case, the MAIDs created in ABM are linked via chain of custody to your business and you have the right and the ability to reclaim them once a staff member leaves. Regular Apple IDs, even those created with your domain, follow the user when the user moves on. The user leaves, the ID goes with them. You have no rights to the ID.
Ah, but MAIDs have limitations. The one that is impacting you is that a MAID cannot have payment information associated to it. No payment info, no participation in stores. Scroll down on this page to Content Availability. But read the entire article to understand the limitation of MAIDs.
You have two options. And I suspect you don't want to hear either. (1) deploy an MDM. Your problems are solved. Well, your app distribution problems are solved at least. I tell all my customers... Deploying an MDM makes everything easy. Deploying an MDM is hard. But it is the right thing to do. (2) Allow users to log into devices with personal Apple IDs. This is a terrible idea because the apps belong to the user, not the company, the personal Apple ID will allow activation lock (Find My) to be associated to the person's personal Apple ID (which means if the employee leaves the device my be left activation locked and bricked unless you go through the proof of ownership process with Apple - or the employee releases the device), and you will likely be granting the employee access to too many features - don't get me started on how large a date leaker iCloud Desktop and Documents is.
Hope this helps. If you need help setting up an MDM, I encourage you to find a qualified consultant by using the Apple Consultants Network Finder.