Hi @PedroNL ,
Based on your description, I think you want to build an app to automatically display all of certain users, then you want to add them to distro group and calendar with certain rights.
Because I don't know the type of your distro group.
I assume that your Group is security Group, you can add them to the Group as below:

// 12717916-8cc5-480d-b0c2-fc3c773f8456 is the group ID.
When I click this add button, I can find ‘ada’ is a member of this group:

If your Group is distribution group, I'm afraid this isn't possible to do in the PowerApps. Because we can only add members to security groups and mail-enabled security groups currently.
Besides, if you want to add these users to Calendar with certain right. I am sorry that we could’t realize it. Because not only the related connector doesn't provide this function, Microsoft also doesn't provide any API to add users to calendars with certain rights, so we can’t use custom connector to realize it.
Finally, you want to build a app, it automate looking at all of certain users via your Azure AD. Please refer to the following steps:
1.You can add Office 365 user connector and Azure AD connector in your apps:

2.Set the OnStart property of APP to:
ClearCollect(allUsers,Filter(Office365Users.SearchUser({top:200}),"yourtenant.onmicrosoft.com" in Mail))
// this formula aims to new a collection which can list all the users in your AAD when you open the app
//because Azure AD connector can only show the specific group member, here I use office 365 user connector.
3.Add a gallery and set the collection as the data source of gallery.
allUsers
// allUsers is the collection I created it before.
4.When you open this app, you will find all of the certain users are displayed in your app:

Hope it helps!
Thanks,
Arrow