After some research, I finally figured out the two different options.
1. MDM to push App Store applications
I made my tests with the AirWatch MDM solution. You can actually push iOS applications to a fleet of devices with Apple accounts connected on them. The Apple accounts connected on your devices can either be manually configured by the users, or you can provide these devices with Apple accounts already signed in.
In the latter case, you need to create the accounts on behalf of your users. You can even prevent them from modifying the connected Apple account. However, I advise against using the same shared account across multiple devices as it's against guidelines. If the account is blocked because of incorrect usage, application updates will be impossible on all devices using this account.
2. Using an Enterprise Program in addition to the Standard Program
In such a case, you can push your in-house applications through MDM not requiring your devices to have a connected Apple account, which is a strong advantage in my opinion. However, if you still want your application to be distributed through the App Store, you will have to maintain two different builds of your applications, with two sets of signing certificates, provisioning profiles, APNS certificates, etc.