Apparently you have to "accept" some sort of invitation or notice from instagram before it will let you link the accounts. We currently do not know where/what/how this process looks like.
After adding users to the list, they will have to accept your invitation before actually becoming sandbox users. To accept the invitation, they can go to the 'Sandbox Invites' button on the top bar of the developer site. In the 'Sandbox Invites' page people can accept or reject invitations, and also stop being sandbox users for a given app.
Note: a user can become a sandbox user of up to 5 apps. You will not be able to invite someone that is already a test user of 5 apps.
Top bar -> Sandbox Invites
Follow the instructions from there, you might have to "register" as a developer or something, but that should get you started.
Instagram integration should be working now.